Firstly, you played the hand fine. Things to be aware of if not already your 3b in sb v utg looks very strong. Therefore I expect BB to have a reasonably strong calling range( this does depend on the player though and is a factor to consider in spots like this)

My estimate of BBs range if a solid player is something like JJ/QQ/AK( ak less likely due to blocker effect) possibly slowplayed aa but unlikely at low stakes. Though this range can stretch as far as AQss/77/88/99/ 1010 for a looser player and therefore we are crushing the board against villains range.
I personally would bet pot closer to a third as it is so dry and we can keep in more hands that might otherwise fold out.
Turn is basically a brick and a great card for our opponent to continue on. I am betting half pot and snap calling the jam.

I am never checking turn as I could lose value in this spot.

Sigh 1010 unlucky ran into the top of the range.

standard cooler brah