On a dry flop if I'm last to act I routinely bet half pot with anything which usually takes it down. OOP I'm more likely to give up.

Since it's a 3-bet pot, which makes it protected to some degree, I'd prefer stabbing the turn after SB checked twice, if that happened. I'd size the stab closer to half pot too.

As played, if SB c/r's, I'd just fold. I think shipping overs is a big mistake. Those kinds of plays will sometimes work but overall they are a cost center not a profit center.