I have experience of caring for a wonderful MIL, she had no assets but another family member has been in a slightly more similar position to your financial situation.
I think, in your position (as I understand it) if it was affordable I would look for a home like one of my relatives which is a 4 bed converted bungalow and has a bathroom and one bedroom downstairs. This would mean MIL could stay with me for short periods but also, if needed, for longer.
This might mean your MILs current pattern of time in her rented home and periods of respite could continue for longer. She would be happy you had moved, she wouldn't go into full time care home immediately, you might even go through a stage of giving up rented home and moving between you and care home.
Only you know if this sounds ridiculous. If your home had been used or intended to be used to support MIL if needed I believe there are slightly different rules about pushing you to sell plus, as you say, you'd have savings available.
One reason I'm suggesting thinking about it this way is because my MIL, having been given 2 years prognosis, survived for nearly 10. Most of that time she remained in her rented home with us or residential carers alternating. HCPs were sometimes bemused that we wanted to do it but she was wonderful and I don't regret it at all.