Live 3/6. Huge pot, Should I have folded AA?

I'm MP1 at a typical 10-handed LAG 3/6 game. Most hands were either limped in (5-6 players) or single raise with callers (4-5+ players). I had AA unimproved in what became a huge pot but at every stage the pot was just too big for me to lay it down. Should I have been able to get away from it at some point?

UTG and UTG+1 limp. I raise with A[img]/images/graemlins/spade.gif[/img]A[img]/images/graemlins/club.gif[/img]. MP2 cold calls. Button reraises. Big blind and limpers call. I cap and everyone calls. $69 pot.

6 players see the flop. Flop is 3[img]/images/graemlins/club.gif[/img]4[img]/images/graemlins/diamond.gif[/img]7[img]/images/graemlins/club.gif[/img].

BB bets. UTG and UTG+1 call. I grit my teeth and raise. MP2 calls. Button reraises, BB caps and everyone calls. Pot is now $141.

Still 6 players. Turn is Q[img]/images/graemlins/heart.gif[/img]

BB bets. UTG, UTG+1, I and MP2 call. Button raises, BB reraises, everyone calls to Button who caps it. Everyone calls. Pot is $285.

River is 2[img]/images/graemlins/spade.gif[/img]. BB goes all in for his last $5 and everyone calls. Final pot is $315.

Results and spoiler comments in white below:
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BB shows 47o for 2 pair.
Button shows QQ for trips and wins the pot.
Other players mucked.

I clearly wasn't going to fold on the river. The only place I think I could've gotten away from the hand is on the turn but the pot was already so big that I was getting odds to hit my 2 outs.

I was somewhat worried about 56 for the straight, but the way the hand played out and my reads on the players meant only the BB could've had 56 and he had proven to be a donk on multiple hands earlier (e.g. raising bottom two pair after the board paired counterfeiting his hand into multiple opponents) so I was willing to pay him off if he was lucky enough to have flopped the nuts. The Button *could've* reraised preflop with 56s but that didn't make sense with the way the table and he had been playing so I discredited it. Everyone else was just calling down given multiple opportunities to raise so it seemed more likely they were on draws than slow playing a made hand.
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