My mom's house was recently cleaned, remodeled, and a professional live-in caregiver was hired. One sibling come and took mom to her house against all practical reasoning. She has never been able to care for her properly. And the sibling with the formal Financial POA cannot convince her to bring her back to her house for proper caregiving. Another sibling with Healthcare PO is her accomplice.
Sorry, but those are the facts. So work it out if you can.
Financial POA should make funds available to the sister who has mom. Saving the cost of a professional live-in caretaker isn't small potatoes. Use some of that money to make sure your sister has what she needs.
And don't forget the main goal: to take good care of mom.
Pam Stegman nailed it, in my opinion. This is a family problem...probably of long-standing. There's really nothing practical you can do, in my opinion. Find a way to make it work. In the end, as long as mom is being taken care of, what difference does it make in whose home she lives?
I agree that it would be ideal if you could find some common ground with your sibs. You state a concern that the sib who took your mother has never been able to care properly for her. You need to keep an eye on this for your mum's sake. Is you mum able or unable to make decisions for herself? You don't mention her health issues, her competence. Your caretaking sib may burn out and be willing to return your mum to her home to be taken care of by professionals. Good luck.