What to do with an overpair against 2 LAGs and a solid player?

[BigNads79]

Live house game 10c/20c - Effective stack sizes ~$55.

I'm UTG with QQ and I raise to $1. 2 folds to MP1 who raises to $4. MP2 and SB both call. I call as well. Pot is now ~$16 and I have $51 behind.

The flop is three small cards rainbow. Everyone checks. The turn is an A. Everyone checks to MP2 who bets $11.

I confess I was really lost here. MP1 and MP2 are both loose and aggressive and their ranges were huge (66+, A10+, high suited connectors, possibly Ax suited). SB was a little loose, but a pretty solid player. His range was more like (88+, AQ+). Against 1 or even both of the LAGs, I maybe make a 1/2 pot sized bet and fold to a reraise (fearing Aces or Kings), but I was also concerned about a check raise from SB, who I expected to have a hand. Both LAGs were also capable of bluff raising, and I figured they knew I didn't have Aces or Kings. My image is TAG, and I'd shown down a couple of rare bluffs.

When the A came out on the turn, I figured someone must have had an Ace, so I checked again...

Any way I could have played this better? I think 3-betting preflop to isolate/define may have been a good idea.

[PietM]

I think this really sucks. I think a big raise preflop could work. But calling and try to dodge overcards is ok too, IMO.

As played: it's a big pot and it's multiway. I think I would bet out on this flop (~$15?). Betting half pot isn't as effective, as it's multiway.
As played, I think your beat badly when the Ace hits the turn. With multiple opponents, I'd check/fold there.