Denial from the White House
Well, the US is publicly denying this.
QUESTION: Thank you, Matt. On – couple of question on Bangladesh. During a recent press conference, Bangladesh Prime Minister Hasina alleged that the U.S. wants to grab the small island of St. Martin. And main opposition BNP wants to sell it to the U.S. That’s why she will be ousted from the power. Last 15 years she is in power without reflection of the will of the people of Bangladesh, though. The AL wants to purchase or grab this small island, or is all this a disinformation spreading by the top regime people? Why it is so important for no reason, the St. Martin Island?
MR MILLER: I will just say that it is not accurate. We respect Bangladesh’s sovereignty, and we have never engaged in any conversations about taking over St. Martin’s Island. We value our partnership with Bangladesh. We strive to bolster our relationship by working together to promote democracy, including by supporting free and fair elections.
The rumours/claims that the less India-aligned parties in Bangladesh (i.e. the BNP) planned to 'sell' St. Martin's to the US have been going on for years if not decades.
Saint Martin's is not big enough
Secondly, even if you don't believe those US denials, and while it's true the US wants [to retain] some base in the Indian Ocean, the Bangladeshi island in question is pretty damn small at just 3 sq km. For comparison, Diego Garcia is 10 times the size (30 sq km), and even that one is considered to be pretty limiting for the US base there.
Saint Martin's location is not convenient
It's also only 8 miles of the coast of Myanmar, and the government of that country is & was typically quite China aligned. So any US base there would be in range of regular artillery from China-friendly forces. (Never mind that with the eternal civil wars in Myanmar--"world's longest ongoing civil war, spanning almost eight decades"-- it's pretty easy to claim some non-state actor did some attack, and the government knows nothing about it. E.g. no one knows for sure who shot at some Bangladeshi resupply boats recently.)
Saint Martin's doesn't have much strategic value
Yeah, it's not entirely impossible to make a base on something like that, see e.g. Minamitorishima (aka Marcus Island -- at just 1.5 sq km), which is a Japanese base these days, but there's not much going on there. Perhaps more compelling is India's militarization of the Andaman & Nicobar islands (but these are much larger and sometimes described as India's "Iron Chain" against China)... which also somewhat benefits the US via a recent deal:
In the Bay of Bengal, the Acquisition and Cross-Servicing Agreement (ACSA) signed in 2016 enables U.S. use of Indian facilities on the Andaman and Nicobar Islands, an “INDOPACOM gateway” for the U.S. This was demonstrated for the first time in October 2020 with U.S. P-8 Poseidon aircraft landing at Port Blair for logistics and refueling support.
The French have obtained a similar deal in 2018 (same source).
It doesn't seem mesh with the current US strategy
True, if we take the Chinese efforts into account, e.g.
at Fiery Cross Reef, which is 274 ha (2.74 sq km) mostly of an artificial construction (over a naturally submerged reef)...
In 2014, the PRC commenced reclamation activity in the area, and it has been converted into an artificial island of 274 hectares (677 acres). There were around 200 Chinese troops on the atoll in late 2014, though this number was likely to have increased significantly in 2015 with the addition of support personnel for the new airbase, including a 3,125-meter-long (1.9 mi) runway and associated early warning radar site.
According to the Center for Strategic and International Studies, it is "the most advanced of China's bases" in the South China Sea's disputed areas, with 12 hardened shelters with retractable roofs for mobile missile launchers already completed. It has enough hangars to accommodate 24 combat aircraft and four larger planes.
Many of those reefs in the South China seas (including that one) are claimed by the US ally Philippines as well. So, surely the US has easier political access to build some such "counter-bases" on islands closer to China if they so desired.
The US has access to "only" 9 bases in the Philippines, including an airbase on the Palawan island and another on Luzon. However limited infrastructure (and perhaps political reluctance) has so far not resulted in any big US bomber deployments there, unlike e.g. 'father back' in Australia.
So, "build bases in their faces" on some previously unmilitarized zones doesn't seem to be the current US strategy, for now.
At least publicly US strategists have mostly dismissed the importance of such small-island bases, in an actual military conflict:
The new bases were sitting ducks, American planners sniffed, and could be taken out quickly in an actual conflict. They are still highly vulnerable, says Andrew Erickson of the US Naval War College.
It all depends on their ability to repel an attack; in case of no air defenses, surely the 'sitting duck' scenario was verified e.g. in the case of an "IS-infested" 1.5-miles long island in the middle of the Tigris River.
Conclusion
The comparative advantage of a small Bangladeshi island (which does not even have an airstrip presently) much closer to the coast is somewhere between unclear and non-existent. I've not been able to find a Western source that even discusses St. Martin's as potentially useful somehow in an anti-China strategy.
