ROFL!! And again, I have to ask, do you even bother watching the show or just make up nonsense on a week to week basis?
Tyrion DID NOT pick Jaime. It was right there in the very first scene, Jaime excused himself before Tryion asked because of how poorly his training was going which is what everyone with a brain has been telling you all along. How is your attention span so short that you can't even follow along through scene 1?
Tyrion: 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 stableboy with my left hand.
Then and only then does Tyrion joke about how amusing it would be to see Jaime fight as the family name would get snuffed out with a single swing of a sword.
Are you OK there, bud? Just watch Tyrion's face. He wanted Jamie to fight for him, it was there in the suggestion. The implication being, at the beginning of the scene, that this is what they were already discussing. It's obvious that Jamie is Tyrion's first pick--any fool can see that. Otherwise, that conversation would not have happened.
The "How funny would it be to see father's face with you on the ground" was another nudge to get Jamie to fight for him. It points out how "useful" it would be as a play against daddy.
Anyway, it didn't happen in the end, and you get to see your Spaniard possibly lose his head now.

(The Mountian looks terrible though--slow and clumsy. Weird that Bronn would stay say fast for his size--when all we saw in this episode was the Mountain being slow and doing nothing but murdering unarmed supplicating peasants.)
But, as this is very important to you, I grant you: