[tardigrade]

Generally when there are no raises to you holding AQo in the BB, then you're holding the best hand and can raise for value. The down side is that with a raise, you're building a pot big enough where you can get trapped in it with nothing but overcards, and you give up a little post-flop deception.

If you think there's any chance you can remove MP1 by making him call 2 cold on the flop, I think you'll want to 3-bet there, especially if MP3 is aggression-prone and will attempt to protect any TPGK with a raise. Even if it is capped back to you, you still have a 2nd best hand with outs to a winner. If MP1 stays in that case, he's paying you EV, and if he folds, it increases your chances of winning.

Given the play to that point, I'd be tempted to bet out on the river, folding to a raise. There's a small chance the second 8 actually helped you by counterfeiting a lesser two pair, and anyone who raises when you wake up on the river likely has you beat badly.