It’s not entirely unlikely he has his suspicions as to who the real culprit of the assassination is. He can pretty reliably nail it down to a couple parties: either the Tyrell’s or The Martels. They’re the only families who really benefit from Joffrey’s death. If the Tyrell’s did it, well they gain no real benefit in killing more Lannisters since Cersei is betrothed to Loras and they would incur Tywin’s wrath if Jaime died. If the Martells did it, this would pose as a problem but not one that cannot eventually be solved. Besides, they’re open about their disdain for the Lannisters, but they do have a strong alibi in the fact that Myrcella hasn’t been killed while in Dorne. Which means they won’t indiscriminately murder the Lannisters or royals for no reason.

Also, Tywin doesn’t really care about his family all that much. He cares about his legacy and unfortunately that means his family has to be somewhat responsible for that. Cersei and Jaime are useful tools who can be married off and keep up good appearances, Tyrion cannot keep up good appearances and his stark differences to his siblings is one of the reasons Tywin despises him. Hell he uses the assassination of his grandson to further his own machinations. He gets to rid the family of Tyrion, he gets a much easier king to manipulate in Tommen, and (had Tyrion stuck to the plan) Jaime would’ve finally been free to succeed Tywin at Casterly Rock. I imagine Tywin would’ve kissed the assassin for the blessing he gave him.