FLOP: When the BB bets the flop, I would tentatively conclude that he does not have a 4 or 33 since the natural play would be to checkraise the flop to trap in the middle guy, or check call the flop and checkraise the turn. When the BB bets the flop I would put him on 3x, a pocket pair or total air. Obviously at this point raising is the right play.

TURN: Once the big blind bets at this point, I would still put him on 3x or a pocket pair most of the time. I would now eliminate the possibility that he has total air, but the probability that the BB has 4x,33 or 88 does go up but its still not high enough for me to be concerned at this point. So the BB does have a hand, and hes probably betting to make sure you dont take a free card if you have two overcards. Given this information, raising is once again the right move. Once the BB doesnt 3 bet you I would be convinced that he doesnt have 4x,33 or 88.

RIVER: This bet on the river is very very confusing. If the BB has a 3 this bet makes no sense, if he has nothing this bet makes no sense, if he has a 4, 33 or 88 why didnt he 3 bet the turn? And lastly if he had JJ and hit a set on the river the more natural play would be to checkraise. The only hand that makes sense to me for the BB to play this way would be J3, and he might be betting the river hoping you have 77,99,TT instead of QQ,KK,AA.

The way I see it, you have showed a tremendous amount of strength so far in this hand, since you have raised all 3 prior streets, and yet the BB is still betting into you on the river, if the BB is not an idiot, he has to know you have an overpair by now, since you have no read on him, I prefer just calling the river over raising the river.

The bottom line is you still dont have the nuts, and since you must call a 3 bet against an unknown IMO if you do raise, then you are laying yourself 2-1 on this river raise. Given all the strength you showed in this hand and the BB is still betting into you on the river I do not think you are a 2-1 favorite in this spot, which is why you should just call IMO.