I also don't like this play at all.

Winning a small pot is still infinitely better than losing a bigger pot by mucking to one final bet.

If he did have KK-TT (using the hand range from a prior post), then I'd be more than happy to win the pot with a turn-raise, rather than allow the Villain a miracle card on the river just to build a bigger pot. Also, we don't know that he has KK-TT, so raising the turn gets us the additional information sooner, and can possibly get some hands to fold. Given the river appeared to be a blank, I'd bet after a check & probably make a crying call if he raised.

If BB had AA or AK-AQ, I still don't like this play. Raising the turn would most likely cause the 3 bets to be in on the turn (certainly with AA .. but maybe just a call with AK-AQ saving you 1 bet from the above posted action), when you still have a live straight redraw to beat him!!! If the river is a blank, you can fold to his bet (or check behind if he's crazy enough to try a check raise), and you still muck having lost the 3 bets of the posted action. If the river is not a blank, you can win a few more from him.

Given you're going to fold to a river 3 bet, I don't the raise after your redraw to beat your most feared hands didn't come.

Perhaps I'm missing some information about the BB, or something else altogether. I don't see it yet.