Having lost his hand, Jamie was not in any position to make a useful fighter. He tells Tyrion as much in the beginning of season 4, episode 7:
Tyrion: Don't give up on me just yet. I survived one trial by combat even though you weren't there to save me.
Jaime: I can't save you this time either. My training has proved that I can't beat a stable boy with my left hand.
Tyrion: Where's your sense of adventure? Even if you lose, imagine the look on father's face when you fall. Our family name snuffed out in the single swing of the sword.
Jaime: It is tempting.
Tyrion: Well, Bronn fought for me once, he'll do it again. If he wins, I expect I'll be in his debt for the rest of my life.
Jaime: If he wins.
I don't know if in-universe this would have necessarily disqualified him from participating as a combatant in a trial by combat, but had he fought, it would have been sufficient enough to embarrass him. Likely, he could not afford to be seen fighting poorly even if the trial by combat were not to the death. Politically, even if he were able to fight, I also suspect he may very well not have wanted to for several reasons such as the poor appearance of a kings guardsman defending an accused king killer. On top of that, the "King Slayer" defending an alleged murderer of the king - not to mention defending his own brother while also being suspected of being his sister's lover. Also, had he even accepted Tyrion's request, he would then have invoked the ire of Cersei and possibly his father as well.
Even if Jaime was set to confront to the Mountain, he is looking for redemption, and dying defending the brother would've been a pretty good altruistic act.
Sure, that would be very altruistic but I'd argue that Jamie is much more interested in living unredeemed than dying for redemption. As for an "altruistic act" of saving his brother, I really don't get the feeling that altruism is a defining goal of Jaime's. Yes, he hates his legacy as "King slayer" but the kind of redemption he is looking for is also mixed up with his sense of pride and entitlement as a Lannister. Consider one of his more vulnerable expositions in the show:
the father and Cersei wouldn't have allowed it, so both the brothers would have survived. Otherwise, they would have chosen a weak adversary to save Jaime and again both brothers would have survived.
Cersei and Tywin are also at odds with each other. Tywin doesn't want any of his children killed: they are Lannisters. Cersei, however, did very much so want Tyrion dead - she blames him for the death of their mother and in general hates him, so she chose the Mountain for combat. Tywin was just as happy with a trial by combat because either Tyrion's fighter loses and Tyrion gets sent to the wall, or Tyrion's fighter wins and Tywin can make use of his son however Tywin sees fit. Despite their differences, I doubt either Cersei or Tywin would have allowed Jaime to participate in the trial by combat at all. The extra manipulations involved in allowing him to fight and also securing an outcome where he does not die seem to me to be too extraneously contrived to be worth the effort when there are plenty of other adequate fighters available.Also, had Cersei and/or Tywin so contrived an outcome, it is just as likely that the political theater would have been so obvious as to further tarnish the reputation of the Lannister family already tarnished by suspicion of incest.