I see no reason to bet on the flop with weaker hands, we will not pay, and we will lose more hands. ranging limp  Call is part of Ax and Jacks, and a flush draw and a gutshot. It should be as cheap as you can walk to the showdown.

I think you played it perfectly. The only possible better line is folding flop vs his raise. I think you have to bet the flop since, if we check back, we're completely in the dark with a bluff-catcher. It's close if you get smooth called, but I'd probably give up completely. And, once the jack pairs on the turn, it's entirely correct to put no more chips in the middle.

Hope this helps