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?