In a word, no - there is no independent verification of this story. Every account I can find ultimately cites back to a single source, the claims of William English.
Some background: Project Grudge, and then Project Blue Book, were the successive code names for the United State's Air Force's systematic study into unidentified flying objects. These generally consisted of obtaining reports of a sighting, encouraging the observer to complete a standardised form, and investigating what could have caused the sighting. The reports have since been unclassified, and you can find the collection online in the Internet Archive: https://archive.org/details/project-blue-book
Under Project Grudge, monthly status reports were started, and these were continued when it became Project Blue Book.
(From Project Grudge Status Report Number 1, online here)
12 such reports were issued, and have been unclassified. They range in date from November 1951 to September 1953 - the original plan to release monthly reports suffered due to the high number of sightings received in 1952, coinciding with articles in Look, Life and the New Yorker, and a highly publicised incident in Washington DC, as mentioned in the 8th report.
(From Project Blue Book Status Report Number 8, online here)
From Status Report 5 onwards, they changed from stating Project Grudge on the title page, as in the first four reports:
to stating Project Blue Book, with Formerly Project Grudge underneath. This would continue up until the final 12th report.
Concurrently with the regular monthly status reports, a special report was carried out where a civilian outfit would use IBM punch cards to enter the data for all recorded sightings up to the end of 1952 for statistical analysis. (This was known as Project Stork).
(The special report being mentioned in status report 8)
(And being mentioned in status report 12)
Ultimately, the final special report was completed and published in May 1955, and was given the name Special Report Number 14:
(You can find it online here.)
There is no mention or record of a report 13 ever being produced. If it were to exist, it is unclear whether it would have been a continuation of the regular monthly summary reports, or another special report as with report 14.
The existence of reports 1-12 and then 14 was well known; in fact, a book was published about report 14 in 1956 (Flying saucers; an analysis of the Air Force project blue book special report no. 14 by Leon Davidson), described here as a "must-have" for UFO buffs.
So, to look at this origin of the linked story in the question. According to sites such as this one and this one, this comes from a report by William English that, in 1977, he was given a document to analyse labelled "Grudge/Blue Book Report No. 13". He recorded his memories of the report onto two cassette tapes. These were transcribed into letters which were circulated amongst the UFO community in the 1980s. Here is the covering explanation that comes with one such example: https://cdn.preterhuman.net/texts/alien.ufo/grudge13.ufo
Text file typed in by UFONET I, original document is 8 neatly handwritten pages of which photocopies have been in circulation in the UFO research community.
Note: What follows is proportedly the recollections of an intelligence
analyst who was given an assignment to prepare an analysis of
the GRUDGE 13 Report. The time frame in which this occurred was
July 1977 with the original GRUDGE report having been published
in 1953. The Grudge 13 report was made up of 624 typed pages on
white paper with a Gray cover. This "Memorandum" as it became
known was first circulated in early 1981.
It seems the letters were then typed up and uploaded to the internet; the uploader from the first website I linked to above being William Cooper, and the second Harvey S. Stewart. Scans of the "8 neatly handwritten pages" can be found here.
In any case, William English's description of the report is:
Publication was withdrawn from pouch. It measured approximately 8" by 11" with gray cover. Heavily bound, paper back style similar to technical manuals. Across the center front it read, "Grudge/Blue Book Report No. 13". It was dated 1953-(1963). In the lower right hand corner was AFSN 2246-3. In upper left hand corner was the word 'annotated'. Across the front upper right hand corner to lower left hand corner was red tape indicating code red security measures. Across the front was stamped in red ink 'Top Secret Need To Know Only Crypto Clearance 14 Required'. Inside front cover upper left hand corner were hand written notations in ink which were blacked out by black felt pen.
This description is clearly different to the other 12 reports - rather than using the standard
Project Blue Book - Report No. XX
Formerly Project Grudge
he states that it said "Grudge/Blue Book Report No. 13".
Rather than being a short 20 or so pages as with the other reports, he claims there were 624 pages, with a full table of contents as follows:
Part 1, “On the design of generators to accomplish strain free
Part 2, "the generation of space time discontinuums, closed, open and
Part 3, ”on the generation of temporary pseudo acceleration locas”
Part 1, Chapter 1 "design criteria for a simple generator and control
system referring to equation 17 appendix A"
Part 2, Chapter 1 "Continuation of Einstein's Theory of Relativity to
Part 3, Chapter 1 "Possible applications of Einstein theory of
relativity at conclusion”.
Part 1, Chapter 2, "Reports of UFO encounters, classification ’Close
Encounters of the 1st Kind' subtitle sightings and witnesses"
Part 2, Chapter 2, ’Close Encounters of the 2nd Kind' subtitle UFO
sightings witnessed within close proximity.
Part 3, Chapter 2, "Close Encounters of the 3rd Kind”, subtitle UFO
encounters and extraterrestrial life forms witnessed and personal
encounters. Subtitle/colonies relocation thereof"
Chapter 3 Part 1, "Military Encounters with UFO’s"
Chapter 3 Part 2, “Military Reports Concerning Sightings on Radar and
Electronic Surveillance of UFO’s.
Subsection 2, Analysis Report, J. Allen Hynek, Lt. Col. Friend.
I find it particularly noteworthy that he claims the report, ostensibly from 1953 with annotations dating to 1963, used the terms "Close Encounters of the 1st, 2nd and 3rd Kind". This scale was created by Josef Hynek and first published in his 1972 book, "The UFO Experience: A Scientific Inquiry." Whilst Hynek was a scientific consultant for Project Blue Book in the 1950s, the scale wasn't created until two decades later. The Air Force files certainly weren't categorised according to a system which didn't yet exist - Hynek himself states:
Air Force files were arranged according to no system whatsoever (not even the simplest cross-indexing of cases, or intercomparison, or classification, was employed)
in his followup book, "The Hynek UFO Report". And looking through the released Blue Book reports we can see the terms aren't used anywhere.
English goes on to recount some of what he claims were the more notable cases from the report that he read - one of which is the Lovette-Cunningham incident. The full text of English's report is:
Report gave a clear indication of reports of human mutilations, most notably was a case witnessed by Air Force personnel in wich an Air Force Sgt. E-6 by the name of Jonathon P. Lovette was observed being taken captive aboard what appeared to be a UFO at the White Sands Missle Test Range in New Mexico. This abduction took place in March of 1956 at about 0300 local and was witnessed by Major William Cunningham of the United States Air Force Missile Test Command near Holloman Air Force Base.
Major Cunningham and Sgt. Lovette were out in a field downrange from the launch sites looking for debris from a missile test when Sgt. Lovette went over the ridge of a small sand dune and was out of sight for a time. Major Cunningham heard Sgt. Lovette scream in what was described as terror or agony. The Major, thinking the Sgt. had been bitten by a snake or something ran over the crest of the dune and saw Sgt. Lovette being dragged into what appeared to him and was descrived as being a silvery disk like object which hovered in the air approximately 15 to 20 feet. Major Cunningham described what appeared to be a long snake-like object which was wrapped around the sargeants legs and was dragging him to the craft. Major Cunningham admittedly froze as the sargeant was dragged inside the disc and observed the disc going up into the sky very quickly. Major Cunningham got on the jeep radio and reported the incident to Missile Control whereupon Millile Control confirmed a radar sighting. Search parties went out into the field looking for Sgt. Lovette. Major Cunninghams report was taken and he was admitted to the White Sands Base Dispensary for observation.
The search for Sgt. Lovette was continued for 3 days at the end of which his nude body was found approximately 10 miles downrange. The body had been mutilated; the tongue had been removed from lower portion of the jaw. An incision had been made just under the tip of the chin and extended all the way back to the esophagus and larynx. He had been emasculated and his eyes had been removed. Also, his anus had been removed and there were comments in the report on the apparent surgical skill of the removal of these items including the genitalia. The report commented that the anus and genitalia had been removed 'as though a plug' which in the case of the anus extended all the way up to the colon. There was no sign of blood within the system. The initial autopsy report confirmed that the system had been completly drained of blood and that there was no vascular collapse due to death by bleeding. Subcomment was added that this was unusual because anybody who dies or has complete loss of blood there is vascular collapse. Also noted was that when the body was found there were a number of dead predatory type birds within the area who apparently had died after trying to partake of the sargeants body. There were a number of extremely grisly black and white photos. From all indications the body had been exposed to the elements for at least a day or two. The New Mexico sun in the desert is extremely hot and dibilitating under normal circumstances.
This seems to be the story which gave rise to all other versions. English adds a few extra details in an interview he gave with a man called John Lear, namely that "Major Cunningham was initially accused of murdering Sgt. Jonathon P. Lovette. Charges were later dropped."
The White Sands Missile Range is a real place, and does border Holloman Air Force Base. However, I can't find any records that people named Major William Cunningham or Air Force Sgt. E-6 Jonathon P. Lovette ever existed. The only Major General William Cunningham I found of the right time period was stationed in the Pacific during the 50s, was army and not air force, and was promoted to Major in the 60s.
Ultimately, I have to conclude that the incident never happened. The only source for it is that of William English - but considering his recounting of the Report has it not match the legitimate reports, and use terms that didn't exist at the time, I believe it has to be a fake.
The linked website also claims that another man, Bill Cooper, saw the same Report 13. However, in Cooper's report, he only states that he saw a document entitled "Project Grudge". He makes no mention of it being labelled Report 13, or mentioning Project Blue Book, and no mention of it containing the story about Lovette.
But eventually, I found myself in possession, holding two documents; one called ‘Project Grudge,’ and another one called ‘Operation Majority.’ Project Grudge contained the history of alien involvement since around 1936, and it began talking about Germany’s involvement with a crashed disk that they had recovered in 1936 and were attempting to duplicate technology... . In Project Grudge I William Cooper saw photographs of these dead aliens [and] of the craft. I saw photographs of live aliens. I saw photographs of autopsies…..of internal organs, I saw photographs of the alien designated ‘E.B.’[or Ebe] who was held in captivity from 1949 until June the 2nd, 1952 when he died. I saw the history of what they had been able to put together [at that time], from incidents in the 1800s that involved aliens and their craft.
As detailed on the site here: https://www.classicalmidi.co.uk/ufo/ufo4.htm Bill Cooper was then one of the people who typed up online John Lear and William English's account of "Grudge 13" and their interview. It states:
This report was typed onto disc verbatim from the original document which was sent by John Lear via Federal Express and arrived on October 8, 1988 by Bill Cooper. I Bill Cooper do swear that no changes were made and do swear that this is an exact verbatim copy of the original.
It seems that, after Bill Cooper saw Bill English's report, the two got in touch - at that point, Bill Cooper claimed that what he had seen was in fact the same "Report 13", as detailed here: http://luforu.org/project-grudgeblue-book-report-13/
Once again, time has proven to be an ally in that evidence has come to light over the past several years that has substantiated everything that I initially said about my viewing of Report # 13. In addition, one other person, William M Cooper, formerly of the United States Naval Intelligence section, CINCPAC, has come forward and admitted viewing an earlier version of the Grudge/Blue Book Report # 13.
~Bill English, 1988
Since Bill Cooper's original claims were just to have seen something called "Project Grudge", and it was only after he read Bill English's report that he switched to claiming they saw the same "Blue Book Report No. 13" I think we can dismiss Bill Cooper's claims as false, which again leads back to Bill English as being the sole source of the story.