I'll preface this as always with limit hands by saying I have little experience with limit.
That said this looks like a pretty obvious check to me. Take a look at the flipside and let's say we do bet, why are we betting? Are we betting for value? What exactly makes us think our hand is good here?. Are we bluffing
? Why are we bluffing for one small bet into a 10 SB pot - what exactly do we expect UTG+1 to be folding? Are we betting 'for information'? What information are we actually getting if villain calls or even check-raises? Are we betting to charge a draw? Apart from QT and T8, there are no draws out there (and a 16% pf raiser isn't capping (edit: limp-3betting)
preflop with either of these hands, right?).
Seems to me there's absolutely no reason to bet, so why bet? Take the free card and see what happens on the turn.
Edit: Just noticed the preflop action, the limp-3bet is pretty strange - has villain tried it before?