Is it permissible to perform Hajj on behalf of a brother who has been banned from travel due to an ongoing political case for 20 years, and who fears death before performing Hajj?

So long as your brother hopes to be permitted to travel for Hajj himself, it is not permissible for him to appoint someone else (to perform it on his behalf); rather, he should be patient and wait. Appointing someone else is for the one who is incapacitated and has despaired of his incapacitation being removed. If he dies before that, there is nothing upon him, and Hajj should be performed on his behalf from his estate, or someone may volunteer to perform Hajj on his behalf.

Ibn Qudamah said: "Whoever meets the conditions for Hajj being obligatory upon him, but is unable to perform it due to an impediment that is despaired of being removed… when he finds someone to act on his behalf in Hajj, and money with which to appoint him, that becomes obligatory upon him." As for "one whose illness is hoped to be removed, and the imprisoned, and the like, he is not allowed to appoint someone else. If he does, it is not sufficient." And your brother falls under the ruling of the imprisoned.