I actually think that the Father Lannister was behind Joff's death. If anything he he hates is someone he ultimately can't control. Joff lacked the control of Cersi. The cunning of Tyrion, and the skill of Jamie(but Jamie was very obedient to their father also) His elimination of Joff and raising Tomnen gives him a good foothold of the position of the King. He still is the king's hand and will wield actual power.
I think he is going to play the trial as Sansa being behind the deaths and bribe the other judges to let Tyrion go. He gets to keep all of his kids that are obedient, Tyrion, Jamie, and Cersi and he gets rid of the only person who was an actual threat to his power which was the King Joffrey. As a bonus he now has Sansa as a fugitive that can be killed on sight.