I still don't get your argument though, say Tywin decides to do what you said. They rig the fight and completely undermine Cersei's accusation. Then what? She resents Tyrion even more, and now she absolutely hates her father. Then what does he do when she goes full on psychopath for not geting her way?
Or, he lets that situation play through, best case scenario: Tyrion wins and he's free, the mountain dies, cersei at least tried but failed, jaime renounces the kingsguard and becomes the heir to the lannisters as promised. (Tywin wins)
worst case: Tyrion loses, is condemned and Tywin allows him to take the black in exchange for Jaime renouncing the kingsguard etc. Cersei gets her vengeance and tywin gets what he wanted out of Jaime. (Tywin wins)
and the actual worst case scenario that you would not even remotely factor in: Tyrion loses, varys and jaime free tyrion, tyrion kills you while you're trying to take a shit thats literally rotting you from the inside. (Tylose)