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If his range is AT, AA, QQ he's actually more likely not to have AT.

6 ways to have AA.
6 ways to have QQ.
4 ways to have AT.

You are ahead 5/8 times. Cap the river, he's gonna raise with any boat and he likes TTTAA too much to call.

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IMO. Bayesian analysis is used too often in situations like these incorrectly. You should use this method of analysis when all of his holdings, when weighted based on previous actions, are equally likely. You can then use Bayesian to see which holding is his most likely due to distribution.

In this case, when he raises the river, I don't think the options are equally weighted.

When he raises after the King falls on the river, I think we can discount QQ heavily. And I think that you can limit his holdings to AA and AT. Without a preflop cap I'm inclined to think that AT is actually favourite; not the other way around.