If the siblings don’t speak to each other, and Dad’s got health issues, (which he won’t discuss with you, as is his perfect right), and he doesn’t want to live with you, and Sis can’t handle the care, I don’t see how this is going to resolve itself, any other way, to be honest.
She’s found a place he has agreed to, and she’s in a better position to know his needs than you, at this point, why upset the applecart? Shouldn’t you be happy they’ve found a solution your Dad’s good with, and leave it at that? As for the finances, it seems most likely his house will be sold to pay for his care expenses.
This family clearly has had, and continues to have, some pretty serious issues, with communication and trust. That being the case, this outcome is not that bad really. No one is being abused, he’s going to a place he helped pick, willingly it seems. His house should pay for it, I’m not seeing the problem beyond no one consulting you about it. Certainly that would have been ideal.
But you say none of them talk to you, your Dad doesn’t want to discuss his health issues with you. And they felt more comfortable making this difficult choice without you in the loop. Perhaps, and I’m just guessing here, it’s because your issues will just make a difficult thing, that much harder for everyone.
Instead of making trouble, consider stepping back, until you have some actual hard evidence that something untoward has occurred. I’m not seeing any in what you’ve told us, to be honest. Beyond getting your nose out of joint, because you didn’t make these choices.