I would 3bet this hand preflop unless the BB is a weak player. We give up a lot of equity with this hand multi way compared with HU. Also winning preflop is a good result and if we get 4bet we don't mind folding this hand. That said I think calling is better than folding and not too much worse than raising.
I think raising the flop and jamming the turn is too aggressive in this situation, the important factor is that the flop is 3 way and not HU. Since it's 3 way the CO should have way less random air hands that cbet - this means that you are perceived to have very few bluffs - this means other players will not continue with worse than KQ very often.
This is a difficult situation to deal with once we reach the turn as all options seem sketchy. IMO it's not that productive to work out the best line but rather to work out how to avoid the situation in future.
In this case I think that 3betting out of the SB more will help you, but by far the most important thing would be to not raise flops with marginal value where you can't have many bluffs (unless you have a very good reason).