Unless the 3 bet will generate a fold pre or on the flop I dont like in this spot. No way of knowing whether it would or not from my shoes but you would know that playing against the villain heads up assuming this was not at the beginning of the session.
The flop sizing is okay to me since that sizing can still be repping 10x and with the 3 bet pre you could be playing a monster that you dont want villain to fold against. However, I think your mistake is actually on the turn. When you check the turn and the board becomes double suited, villain should know there is no way you could have anything that strong since if you did you would want to protect it a bit, get value from it, and charge the potential draws that villain could have. Villain could put together now that you have air or have some small pocket pair at best and will much more inclined now to call down on the river. As played going to the river (this would actually could make you lose more in this hand then you did unless they check behind) I would go with check call as the play and hope villain bluffs at this pot thinking that we are not that strong. Villain could
bluff with the scare card river here and generally probably would check behind with their 44-88 (less the 55). By checking you might save some money against small pocket pairs and gain more from villains air since villain probably folds their garbage to your small river bet.
