Hitting bottom set on the turn

[Post-Oak]

2/4 NL ($400 max) I ($386) am viewed as tight, BB ($526) is not terrible but I would not call him solid (see further description below). I don't know enough about the short stacked UTG+1 ($132) to have an opinion, but his name is ICEMAN5 and I was thinking it could be the poster who posts here under the same exact name. Given that he horribly misplayed this hand, I don't think it could be the same player. Of course, I did not know this until he showed his cards.

I'm sitting UTG+4 and am dealt 4h 4c. The preflop action went like this: UTG+2 limps, UTG+3 limps, UTG+4 makes it 18 to go, I call, BB calls, UTG+2 calls, UTG+3 calls. 5 see the flop.

Flop (pot = $92):
6s 7h 9c

Checked around.

Turn ($92):
4s

BB checks. UTG+1 bets $11. Folded to me and I raise to $105. BB raises all-in. He has us both covered. I have $263 left and UTG+1 has $103. UTG+1 calls. There is now $574 in the pot and I have $263 left.

I see the BB as a bit of a weak player. His problem is he can't lay down hands like 2 pair, even when someone bets big on the river and the board is scary. I've seen him call off a lot of money this way. I also have a note that he got it all-in with QQ preflop against a good player (and he was the caller). So I don't have too much respect for his game, but this all-in check raise is not something I would consider normal if he had 2 pair. He tends to lose money by calling with those hands, but not being insanely aggressive with them.

Thoughts? I will give my thought process with the results.