Is it permissible for a married man who buys the Eid sacrifice to include his needy brother in the intention of the sacrifice, knowing that he is the one who pays the full price, and that his brother suffers from a chronic mental illness that affects his choice of the sacrifice, and to whom does the reward of this sacrifice return?

Your sacrifice is not sufficient for your brother as long as you do not spend on him, because one of the conditions for the permissibility of sharing in the sacrifice is that the sacrificer must be spending on whoever he wishes to include in his sacrifice. As for your brother's chronic mental illness, it does not prevent the validity of the sacrifice if he performs it; rather, the sacrifice performed with the wealth of a mad person is valid.