An interesting question. I don’t think Tywin thought much of Aegon V as king – for one thing, their political projects seem completely opposed and I think you’re right about the kids. Also, I think Tywin probably resented the way in which Aegon kept intervening in the West because of Tytos’ weakness, such that Aegon was a reminder of how far the Lannisters had fallen.
But this leads me to wonder – who did and didn’t support Aegon V? There were clearly rebellions in the Westerlands and Stormlands, and I wouldn’t be at all surprised if the Brackens rebelled what with a Blackwood as the Queen. I could see the Reach as also a potential area for rebellion, given the marriage crisis and the Reach’s more heavy emphasis on chivalry and social caste.
I’m guessing Aegon V found a warmer reception in Dorne, both because he had both Martell and Dayne blood and because of the Dornish belief in collective social responsibility. I wonder if a young and idealistic Jon Arryn approved or disapproved?
I would be curious to know what Jon Arryn thought of Aegon V’s decisions, for good or ill, since I seem to recall from somewhere in TWOIAF that he was one of the lords who openly disapproved of Tywin’s actions against the Reynes and Tarbecks (although if I’m wrong, please tell me).
WOIAF doesn’t say which lords disapproved. I’ve suggested that Jon Arryn might be one of them, but there isn’t any direct evidence.