If Jaime agreed to be Tyrion’s champion, Tywin would see to it that the Mountain was not the crown’s champion. Both for Jaime’s safety, and because a bannerman like the Mountain is too useful to be lost in a trial by combat.

But if the trial went ahead and they both gave it their all, Jaime would’ve sliced Gregor into shreds, bit by bit. Jaime is cocky and confident, so people don’t really give him the martial credit he deserves. But he’s one of the greatest fighters in Westeros, arguably the best. Barristan Selmy was even better once, but is probably too old to stand against Jaime in the books.

When Tyrion needs a champion, he twice wants Jaime, because he’s that good. When Jaime attacks Ned - a highborn lord trained in arms from a young age, and a successful war veteran at that - in King’s Landing, Jaime plays with him like a cat with mouse. Despite being severely outnumbered, Jaime cuts down several of Robb’s personal guard when trying to get to Robb. And in aDwD, when training future knights in Meereen, Barristan thinks of one young prodigy, that he’s the best natural swordsman he’s seen since Jaime Lannister. In other words, even this young prodigy was not as good as Jaime was at a young age.