How does your sister in law actually feel it about this, herself? You said she doesnt' want to move, but has she told her mother that directly or just told you and your husband? She needs to advocate for herself, and strongly but maturely, with her mother. If your husband and you can help her practice what to say, so she's respectful but firm, and so she has a written list of reasons why she should remain at her school for this last year, that could help. She might even want to write her mother a letter laying out why she should stay, and offering some options, such as staying with you if that's possible, etc.
I agree that it's your husband's place to discuss this with his mom. And I think he should for his sister's sake, discuss it.
His sister is doing well academically, has friends and participates in sports (and probably other extracurriculars too?). Uprooting her now is upsetting, and while yes, she can live through that upset and kids do it all the time -- one argument for keeping her where she is, is that if she's going to college, having to restart at a new school means she's back to zero in terms of getting into extracurriculars, getting onto a sports team that's already pretty established, possibly seeing some negative impact on her grades if the new school has a very different curriculum or style....
While I would feel the larger problem is "mom should wait a year, ensure that new boyfriend is the real deal and willing to wait for her," I would not say that to mom; but your husband could try arguing that a college-bound HS senior should stay where she is already successful. Even if sister is already accepted to a college, they do review senior year grades and if hers are affected it could affect the offer. Colleges also like to see a lot of participation in leadership roles and extracurriculars etc. these days, and having to reboot for just one school year is going to make it hard to keep up those kinds of things. Would mom be swayed by any such arguments?
Really, the true issue for me would be forcing the girl to give up all that's familiar for what will be a temporary, one-year change for her, if she is heading off to college after HS. Staying put makes so much sense that I agree with you that mom isn't thinking rationally -- but she is not going to see it that way and will probably get defensive if your husband puts it like that.