It is too early to tell. We do not know its origin. We have neither any evidence for nor anything against this claim.
1. First of all: While the question 'as is' misrepresents the origin of the claim:
The Washington Times article just says this:
Coronavirus may have originated in lab linked to China's biowarfare program
That does not say 'the lab made this'! It also does not say that this is a lab developing bio-weapons. The article just states that
the institute is linked to Beijing’s covert bio-weapons program.
What seems to be known now via genomic sequencing is:
Developing a vaccine for example — whether for 'normal' viral diseases or even for something someone might want to try as a bio-weapon — is not the same as 'making' a weaponised virus. Especially not 'constructing it artificially' or even breeding it up to increased virulence…
2. Second, we have limited knowledge about this, and that state is likely to continue for quite some time.
Given what’s known about the pace at which viral genomes mutate, if nCoV had been circulating in humans since significantly before the first case was reported on Dec. 8, the 24 genomes would differ more. Applying ballpark rates of viral evolution, Rambaut estimates that the Adam (or Eve) virus from which all others are descended first appeared no earlier than Oct. 30, 2019, and no later than Nov. 29.
The progenitor virus itself was almost certainly one that circulates harmlessly in bats (as SARS does) but has an “intermediate reservoir” in one or more animals that come into contact with people, Andersen said. Presumably, that reservoir is one of the species of animals at the Wuhan market thought to be ground zero for the outbreak. The ancestor of 2019-nCoV existed in that species for some unknown time, never infecting people, until by chance a single virus acquired a mutation that made it capable of jumping into and infecting humans.
The genome sequences suggest that was a one-time-only jump. “The genomes [from the 24 samples] are very uniform,” Andersen said. “If there had been multiple introductions,” including from many different animals, “there would be more genomic diversity. This was a single introduction.”
— Sharon Begley: "DNA sleuths read the coronavirus genome, tracing its origins and looking for dangerous mutations", StatNews, Jan 24, 2020
Whether the Chinese lab in that region is linked to any bio weapons programme is of course a boon for any conspiracy theory. But this is wholly irrelevant to whether the lab is the source of that outbreak now. Even if there are zero links to 'the military': a lab that studies viruses can leak those viruses.
And that such leak might have occurred is deemed as plausible as the wholesome naturally infected via animal reservoir on the market place or in kitchen.
Conveniently, the Wuhan National Biosafety Laboratory in the centre of the disease outbreak is officially equipped to handle the virus, 'safely':
— Nicoletta Lanese: "Only one lab in China can safely handle the new coronavirus", Live Science, 22 Jan 2020.
Other experts other than the source quoted in the Washington Times article were also not overly excited about that lab:
But worries surround the Chinese lab, too. The SARS virus has escaped from high-level containment facilities in Beijing multiple times, notes Richard Ebright, a molecular biologist at Rutgers University in Piscataway, New Jersey. Tim Trevan, founder of CHROME Biosafety and Biosecurity Consulting in Damascus, Maryland, says that an open culture is important to keeping BSL-4 labs safe, and he questions how easy this will be in China, where society emphasizes hierarchy. “Diversity of viewpoint, flat structures where everyone feels free to speak up and openness of information are important,” he says.
The plan to expand into a network heightens such concerns. One BSL-4 lab in Harbin is already awaiting accreditation; the next two are expected to be in Beijing and Kunming, the latter focused on using monkey models to study disease.
Lina says that China’s size justifies this scale, and that the opportunity to combine BSL-4 research with an abundance of research monkeys — Chinese researchers face less red tape than those in the West when it comes to research on primates — could be powerful. “If you want to test vaccines or antivirals, you need a non-human primate model,” says Lina.
But Ebright is not convinced of the need for more than one BSL-4 lab in mainland China. He suspects that the expansion there is a reaction to the networks in the United States and Europe, which he says are also unwarranted. He adds that governments will assume that such excess capacity is for the potential development of bioweapons.
“These facilities are inherently dual use,” he says. The prospect of ramping up opportunities to inject monkeys with pathogens also worries, rather than excites, him: “They can run, they can scratch, they can bite.”
Trevan says China’s investment in a BSL-4 lab may, above all, be a way to prove to the world that the nation is competitive. “It is a big status symbol in biology,” he says, “whether it’s a need or not.”
— David Cyranoski: "Inside the Chinese lab poised to study world's most dangerous pathogens", Nature, 22 February 2017.
This is even echoed – again – in a recent newspaper article from April 14, 2020, this time in the Washington Post: "State Department cables warned of safety issues at Wuhan lab studying bat coronaviruses"
That it is a genuine zoonosis was the immediate reaction in expert circles. That it really is 'that' is nowhere confirmed as of now. We know that the Virus was originating from Wuhan, that at least one lab working in that field is there. And that the Wuhan National Laboratory is 'declared safe'.
Not more. Accidents and other incidents can happen, and did. Whether one did at this facility remains speculation, but one that is neither proven nor disproven at this time.
Stating anything as conclusive on this question seems to be another case of China coronavirus: Misinformation spreads online about origin and scale. So please, hold your horses.
— Kat Eschner: "We’re still not sure where the Wuhan coronavirus really came from. China’s wet markets don’t tell the whole story", PopSci, January 28, 2020.
The earliest cases now analysed say that out of 41 patients 13 had no link to the sea food/wet market.
Exposure history to the Huanan Seafood Wholesale market served as an important clue at the early stage, yet its value has decreased as more secondary and tertiary cases have appeared.
Many important questions remain unanswered, including its origin, extent, and duration of transmission in humans, ability to infect other animal hosts, and the spectrum and pathogenesis of human infections.
— Chen Wang: "A novel coronavirus outbreak of global health concern", The Lancet, Published: January 24, 2020.
So it's prudent to remember that initially the MERS virus was shown to originate in a patient zero from June 2012 in Arabia, while later analysis of blood samples traced it back to an unidentified hospital outbreak in Jordan in April 2012.
We have to wait for more information.