Often, taking care of a loved one is a terrible strain for the other. We saw one almost die himself taking care of his wife thinking he was doing his best.
There are a couple of strategies. If your mom has flown on an airplane, remind her of the safety talk and how you are reminded to take care of your self before you take care of others. This is not so much selfish, but if she is not around to take care of herself, she cannot help him at all. It is also safer where doctors can visit him and she does not have to take him to doctors etc.
Make an appointment with a trusted and compassionate doctor to explain to her how the cost of taking care of him is costing her her health and that while the family loves you both, his disease is simply going to progress and he needs to be in a safe place. It does not mean that anyone loves him any less, but sometimes in a structured and monitored environment, he may actually improve a bit, she can still visit any time and it can help both of them.
His reluctance to go is very normal. She should work with the social worker at the place about how to work with this. We had a very hostile father who did not want to go in at all. In one place he was kicked out for holding a knife from the lunch tray and it was found in his bed - I guess he was going to make a break for it. We hunted and found a nice place. It was not an easy transition but he finally did and he improved. We had family and friends who were critical of this decision but he kept accusing of us of stealing and I am disabled, so we could not take him in our home, and he called the police on his care givers so often they quit, so we were determined to find him a good place.
I visited him often and this is quite important. He did adjust and improve with the schedule.
It is never easy - but in the end, it will be better if the place is good.
So sorry your going threw this along with your family
It's not easy to do, my mom passed from this and between me and my Dad we keep mom at home, very hard job and we are much younger than your grandmother
But if she's not wilng than maybe ya need to give it a little more time, can ya hire someone to help out with him in the home
If he's getting worse than soon he won't put up a fuss and it will be easier for grandma to put him there
Prayers for all of this family