I would say it is a clear fold almost always. You are often dominated (by both players) and do not have a lot of implied odds, so calling is a bad option. The tight reg should have a very strong range when he flats on the bu (probably 88+, Aq+, Kqs, Ajs) and probably does not 4bet with Aa, Kk.
4betting could be an option, if you know your opponents fold enough of the time. But there are much better bluff 4betting spots.

KQs, AJs+, AK and pocket pairs I would call, AA and KK 4bet to val.

But I would not know what to do with AQo in this situation. With this hand, i think all 3 options have merit.

Does your answer change if button wasn't in the picture?

I would tend to continue with more TP mining hands like ATs and KJs, AQo is against a lag always a call and I would fold the bad pocket pairs (I would only continue 88+).
KQo is still too bad to continue I think.
And I would 4bet light (2.2x) with AA, KK, some AK and some Axs.

Yea I want to fold here as well for reasons already stated. It's a good price and we are deep so it's not horrible but it's gonna be a loser in the long run. Being OOP and considering the ranges there's not much we can do but fold. I'm more inclined to defend with KQs as the suitedness gives is that much more playability. Still even I find that to be marginal.