Difficult situation OP.
As a mum of a 10 and 12 yr old boy, I'd say at this age they need a surprising amount of love and affection. People are not generally "soppy" about 11 year old boys, and they can eb challenging, but basically need a lot of love and hugs and reassurance.
Pretending to fall asleep when younger sib wants to play a game is not "kicking off" or even being horrid. that is normal sibling behaviour.
I wonder if your expectations of his behaviour are simply unrealistic? When we have little cousins over, the boys play with them for an hour or even 2, but definitely not all day.
Can you give him praise when he is being nice and helpful, and calmly correct when he is being unkind?
It is very important for boys this age to have a good relationship with their dad, so maybe they can go and do stuff the two of them ? Go out for the day? Mine do things like fishing, or big walks, or cinema/bowling (kids am cinema at Vue is cheap, if there is one near). That would give you a bit of space as well.
It must be hard all of you in such a small space!