Sometimes being obstinate pays off.
I just got a call from Catawba -- they found some 'discharge assistance money' and a facility with a good track record working with the patients. No, it's not assisted living, it's a nursing home with stable and trained staff. I've had other residents there (and an elderly relative who was there many years ago) and have a lot of respect for the owner and staff.
Unfortunately the underlying problem still exists: Auxiliary Grant is still a cruel joke. The result I got is in part because I had the power of a court order granting me authority. Many people don't, and don't know how to argue.