Many people with dementia improve when they go into a nursing home - and it sounds as though a nursing rather than a Care home would be more suitable for your father - so please don't discount the idea. Your mother would still be caring but just not so hands-on and she will be able to have some rest. She can still visit every day, I did.
I don't think you can compare the care in the home with that in a residential situation.
I hope for the sake of her health that your mother will consider it.
Yes, it costs but your mother will be left with some money and the house and her savings will not be affected.
I have not heard that people improve in a care home. Is that true? When he was in hospital, in a rehab ward, the talk was that patients improved once they were home. And my father did! But, In a cascade of events, he had a grade 1 pressure sore, (we kept him on his chair too long!) and the district nurse said bed-only until his mark was invisible to the eye. So then he was a week or two in bed, the turner, a distant memory, and now he is moved with a hoist to his chair for 3 hours a day, which makes him more awake and alive. Bed-bound is slow death. We are selfishly awaiting news of an air, electric, cushion for his chair, and then, more time out with us.