Moxie Mike
Full House
You could have improved to the best hand so betting doesn't immediately sound like a bad choice. The problem with betting though is you have no plan for the river if you are called and you're OOP. It's reasonable that someone has a wheel or a set and checked it back... a pretty common play in PLO in multiway pots while holding non-nutted hands. Against 3 opponents that seems plausable.I check and it actually checks around.
Turn:.
Action on us...
What are you representing by potting it in this spot? That you were sandbagging a made hand on the flop? That doesn't seem believable after raising pre from the BB. You could be repping that you spiked a set on the turn... but your hand still is problematic if you're called.
Depending on who's in the game, you also run the risk of someone in position floating. Your hand really can't take any pressure, and you're going to tell them that by checking the river. A savvy opponent could easily move you off your hand by potting the river when checked to.
So the only case for potting it here is if you're also going to pot it on the river. I don't think a two-barrel bluff gets through if you're called on the turn. Betting = -EV.
So checking is the better option. If someone in position takes a stab at it, you can possibly call and see what they do on the river. Folding if someone bets isn't a bad play either.