Honestly if the child's biological mom is pushing this stuff you're gonna have a huge uphill battle, probably impossible, to convince her. The best you can do is show your husband stuff. I find myself returning to this page again and again: UCSF Hospital Guidelines for Trans Youth It's a bit of a long read, but you'll notice that they repeatedly mention the lack of evidence for the treatments efficacy while they are urging for them. The UK NHS has recently updated their guidelines too, here is an article that summarizes the changes, they've gone from being super gung-ho to much more conservative about this stuff. Note: The article itself has a clear editorial bias, but the content they've drawn from the former and current NHS websites word for word does not.

I think that's an important bit: even the most pro "transing kids" institutions concede that evidence is unclear or insufficient. You can't argue that major medical institutions in the most progressive pro-trans parts of the world are biased against trans kids.They already express support as strongly as they can while still having to admit there isn't actually good evidence. You won't convince someone by showing them a source that already is primed against trans people, show them how neutral sources are also admitting the same issues.

The case of James/"Luna" Younger might be of interest to you guys if you're in the USA. Link to article with details. Basically, a male child's mom thought he was a girl and the dad disagreed and they divorced and had a long drawn out legal battle over the child's treatment. The conclusion was ultimately that a neutral third party would make medical decisions if the two parents couldn't come to an agreement, and that currently no treatment (like hormones or surgeries or what have you) was being forced or recommended. The court also found that neither parent was abusive. The dad is not going to be forced to call the boy "Luna" or use female pronouns. If your husband's ex really starts going gung ho about the trans thing, that might be the route your husband will have to go down (taking legal action).

Ultimately though, if this is a very new thing, I don't think there's too much to worry about. Maybe an unpopular opinion here but I don't care about social transition for kids, like if they want to change their name and wear dresses then go nuts. I'd have a hard time using different pronouns but to me the bottom line is that as long as no medical interventions are taken, then it's just another phase. Like if you had a teen child claiming to be bisexual when you knew deep down they were straight, you'd let them experiment and learn for themselves right? The real issue is the puberty blockers, cross sex hormones, and surgeries. Those medical treatments mean PERMANENT lifelong changes. If any of those are on the table, that's the time when hard discussions up to and including legal action may be necessary.

For puberty blockers in particular, you can google "Lupron", one of the most common kinds. There are TONS of lawsuits against the maker of Lupron that come mostly from women who as children were prescribed lupron for precocious puberty. Now realize those women just delayed puberty a few years but ultimately went through their biological puberty, and imagine how much worse the possibilities might be for kids that take cross sex hormones (which do have tons of documented side effects) on top of that.

Another case that might be "fun" to dig into is Jazz Jennings, a MTF trans youth whose entire life is on a TV show on TLC. From that show we know that the puberty blockers + hormones combo meant that he never went through male puberty (so he was a teenager with a child's penis), and when he got the genital surgery, there was not enough penile tissue to form a vaginoplasty, so they had to take tissue from elsewhere in the body (somewhere in the stomach I believe?) and he had major complications and his new "vagina" somehow collapsed. He also has never felt sexual attraction or an orgasm, is permanently sterile, and has major depression. And Jazz is the one they hold up as the model case of a "successful and thriving trans kid". There are many clips from the show on YouTube where you can learn more.