Well this is a dilemma for sure. On one hand there is the problems between you and your Brother and the need to find help for your Dad. So try to forget your Brother and his attitude and concentrate on your Dad. You say your Dad gets up, sits in his chair and watches TV with no idea what he is watching; that sounds a bit conflicting to your earlier statement that he is aware of what is going on around him and is very 'with it'. As you have doubts about your Dad's care start the ball rolling to at least have some respite; he may have forgotten what happened the last time he went to a CH. You could 'sell' the idea as the Doctor wanting him to get his legs working again. 'Love lies' are needed at times.
In a CH/NH your Dad could quite easily get more stimulation-and even settle in well.You won't know until you try. One more point; I firmly believe that Dementia sufferers are sensitive to atmosphere's-happy or tense. Your Dad may be picking up on the stress between you and your Brother-in a CH/NH he would avoid all that.
I feel sorry for you that you will probably have to be the one to make the decision with no input from your Brother, but have you considered that your Brother may be suffering from depression? I remember shouting at my Husband to 'get in the bleep, bleep bath' after I had cleaned the hall carpet for over 2 hours in the middle of the night only to turn round and see him messing himself and walking through it again. Then I found he had rubbed his own waste on the dogs ears! Was I unhappy with myself? Was I ashamed? Was I terribly, terribly depressed-yes to all of that. I didn't want to react but the tiredness that invaded my every being didn't stop me being hateful to my poor Husband. Caring 24/7 is absolutely bone shattering, all engulfing hell. Your Brother may just feel the same.
Lyn T