I think the key thing is to have the conversations at the right time, in the right context. Like, the best way to avoid saying "I support trans rights but [valid tangential belief]" is to separate out those two clauses so that the valid tangential belief isn't appearing to undermine one's support for trans rights in general. Like, even rewriting the sentence so that it is two separate sentiments without the "but" can make a big difference.

A tangential analogy that comes to mind (and I want to emphasise that I don't think your example comes anywhere close to this kind of behaviour) is when assholeish "Men's Rights Activists" (MRA) will complain about feminists not caring about men's suffering, but then be suspiciously quiet whenever feminists try to involve them in the conversation (I say this as a feminist who is passionate about understanding the ways that men also suffer under the patriarchy, and who has become a much more effective feminist through solidarity with men).

But then when a feminist (or any woman, really) starts to talk about things that has affected them as a woman (or someone perceived as a woman), then suddenly the MRAs will jump in to shout over whoever is speaking. It makes it blatantly obvious that they don't actually give a fuck about men's struggles, but just want to derail the conversation and suppress women's voices.

Like I said above, this is a completely different scenario than the hypothetical you described, but they're similar in that the more appropriate response in both cases is to show some grace and make sure we discuss our issues in the space and context that's most appropriate. Like, if the phrase "I support trans rights" is being said, then the conversation is probably about broader issues, and is likely to involve some people who don't support trans rights. Discussing issues like animal derived medication is probably more suited to an environment where everyone there is either trans, or a trans ally — and not just because of the harmful effects of the "I support trans rights but..." framing, but because if someone cares about whether people use animal derived medications, then the last thing one would want is for that point to be hijacked by anti-trans activists who don't give a fuck about animal derived medications