Nah they can just ignore the law.

Seriously though, I'd guess their funding comes from the government in some way, and if they do not adhere to the laws governing that, their funding could be completely pulled.

Edit: To be more clear, it's likely the definition of woman in the eyes of the govt would determine what aid groups do with funds provided by the government if they say they serve that population, so it could be considered defrauding the govt if you're helping trans women with funds allocated only for "women" under the new definition.

I think it's fucked up the court ruled this way, but it's at least understandable why this institute would need to follow along unless they want to try to self-fund in some way, which might not work at all and then the good they are doing is completely lost. They also did say they're going to work on it from a legal perspective so unless that's just BS (could be) then it seems they are trying to do the right thing and advocate for trans rights.