What would Tywin do?

So, I wanted to work on a Lannister fic with a Lannister OC being the main character in it, and I needed a bit of advice on what exactly Tywin would do with the actions I have planned for him.

Gerion Lannister, the middle child of Tywin Lannister, a year younger than Jaime and Cersei. He is proud, rebellious, skilled, very intelligent, and can be considered reckless. When Jaime is named to the Kingsguard at the Tourney, shortly after Gerion is betrothed to Lysa Tully.

Note, Gerion is friends with Oberyn and is rather friendly with the Dornish in particular, this explains actions in the plot.

The story I have planned for it in quick summery is, Gerion ends up saying "I'd sooner feed myself to the lions" to Tywin and fucks off to Essos AFTER Tywin already made plans with Hoster, (this creates an early rift between the Lannister's and Tully's). Gerion was secretly in contact with Oberyn and he ends up meeting with Oberyn and they travel together around Essos, this is before the rebellion.

The planned return of him is him being at the tavern that Catelyn confronts Tyrion in, he obviously stands up for his brother and due to this, he and his wife gets apprehended as well.

The thing here is, while all this is planned out (his being in Essos, honing his craft, gaining a unique fighting style, chilling with Oberyn, marrying a Lyseni woman of noble blood, and returning to Westeros at the wrong place and wrong time with his wife). I do not know what Tywin would be doing in this situation.

Here, he has three sons, two of which are golden heirs. But both are snatched from him, Gerion fucking off to Essos because he didn't like the betrothal, and Jaime joining the Kingsguard. I'm not sure of the actions he would take with Gerion in Essos. If the planned story above could occur realistically in the setting.