We live in a remote rural area and we have been offered 3 hrs a week of a home health aide. We tried it before and it didn't work because dad needs a companion to socialize and to protect him from memory problems. Home health aides are more for specific medical or hygiene needs. The time spent scheduling an aide without meeting them to see if dad will be able to talk to him, the time spent isn't worth having someone who may not be any help. What is needed is someone who will come less often for a longer time. The last one we ended up helping him more than vice versa. How can we decline this while still maintaining the very real need we have for dad's caregiver to get some time off AND dad to talk to someone besides just his caregiver? I think the only option of choice is to hire someone out of pocket. We are in NJ and he needs aide and attendance I guess but don't have it yet.
Dad's need for a companion is a separate issue and should be addressed as something in addition to the aide. Is adult daycare available in your area, even once or twice a week? Some volunteer groups provide visiting companions, have you asked about that possibility? You don't say what your dad's needs are, but perhaps you should start planning for the long term and looking at his relocating to an independent living or assisted living situation where he would not be so isolated and would be able to socialize with his peers. I personally wish I had insisted that my mother make the move instead of allowing her to remain isolated on her own, I'm sure she would have been much happier and less lonely.
The fact that the caregiver is resistant to help seems to be the real problem, and one that will be hard to surmount. I expect that no matter what solution you come up with they will find fault with it. There are no perfect solutions and we have to be willing to work with what we have, if the caregiver would prefer to play the martyr it becomes very difficult.