I would start the flop with a check, especially multiway. The 3bet cold caller has a decent amount of made hands. BTN as well. Our advantage from AA, KK, QQ becomes smaller, especially when BB would continue with a very defined range.
Multiway I just check very often towards IP and defend a lot with checkraise.

Cant agree more with IIICitanul..

However, as played, I dont mind betting either smaller or checking.. again it would be exploitative, but I think that on this card, our bet will generate lot of fold equity, so that we have +ev bet with anytwo. That said, I prob like more x/calling, giving him the opportunity to bluff his QTs,Q9s,T9s, but also bet for protection his Kx.

More info on villain would allow us to make more precise decision. Against a tight reg, I am betting smaller OTT, vs agg reg I am gonna x/c and vs a station I would use similar sizing as you did (to still keep alive his weaker pairs).

But I wouldnt be in this spot since I am also checking this flop (prob x/raising, depends on the sizing)