I'm pretty sure there is some legal reason that a Kingsguard cannot represent the King's accused killer in trial by combat.
IIRC, this is correct. I believe Jaime was Tyrion's first choice, but he was prohibited from championing for him. However, I don't remember why Bronn wouldn't/ couldn't do it.
I don't think Bronn is in King's landing/and or they can't find him.
The quote about Bronn is correct, he refuses because someone offers him a better deal... IIRC he even asks Tyrion "Can you beat a lordship? [in reference to his earlier quote "If anyone asks you to betray me, whatever they're offering, i'll beat it!"]" To which he begrudgingly and without avarice acknowledges that he cannot.
However, in the books, Jaime was still on the road when Joff was murdered, and Tyrion is in chains when he returns to Kings Landing. He doesn't reveal his presence to Tyrion, and only observes his trial from afar. So Tyrion, both doesn't know hes available to call on, nor does he know of his disfigurement (no one tells him anything whilst in the cell, unlike the show he's kept in isolation) so he wouldn't even have suggested calling on Jaime to act as his champion.
As such, in the show Tyrion knows that Jaime is alive and availible and most importantly, not wanting to give the impression that he's lost the ability to fight... IMO, this sets up a plot twist that could be very in keeping with the spirit of ASOIAF and revealing about the characters involved. I.E - Tyrion hates Jaime for ever not because of his youthful betrayal, but for abandoning him as his champion.