Flop:
I think BB could be doing this with 1) a bluff at a dry board, 2) jacks, 3) a set. Versus bluffs you want to call and let BB keep firing. Versus jacks, you may want to raise, but there's a limit to how many streets of value you can get if you do. Versus sets, you want to pot control.
Assuming a reasonable preflop calling range, BB will make jacks about 13% of the time, and sets about 5% of the time. But I think a lot of players would take a check-call line with jacks, so ultimately I'd say that BB has about the same amount of jacks as he does sets in his range. Given that you'd lose a lot more to sets than you would win from jacks, I would flat. Especially given that it helps you extract maximum value from bluffs.
Turn:
I wouldn't expect jacks to fire on the king turn, but bluffs and sets would. Maybe some bluffs that caught a king too. Which makes your hand a bluff catcher! Which means the decision here depends on how often you expect your opponent to bluff.
It's tough to say without any more information, but I lean towards folding. I'm not sure how many live $1/$3 players are two barrel bluffing much, especially without too many draws on the board. And a good chunk of the bluffs probably caught a king.