I think you have to bet the flop. The fact that you won’t have straights is actually all the more reason to bet, because this will be one of the better hands for continuing to a raise. If it’s true that you’ll be unhappy about it (in other words, that the EV of not-folding will be small), then you also won’t get raised that often. Even with so many people seeing the flop, the chances aren’t THAT good that anyone has a monster hand, especially as hands as strong as two-pair might not get raised. The more often you get raised, the more you’re getting raised by weak hands, which increases the value of continuing.

Another way to think about it is this: your hand is way too good to check-fold. Your range is too weak to check-raise (this is the most significant implication of the your being capped here). So your options are check-call or bet. Wouldn’t you rather bet and take whatever fold equity you can get?