well played
Betting flop is alright too as x/R is alright too. When you bet and hit flush you can go 3 streets, when you bet and hit straight you are protected and have 1 street, when you bet and everything bricks out you in bluffcatching mode

What happened to the river? :)
Also you have so much equity on the flop but there's a fair few bad turn cards for you, you're multiway so your hand is more vulnerable than if you were heads up. Why check? Button's range is very broadway heavy, you're going to see a majority of check backs there especially multiway. I would always bet that flop with this hand. I don't know the equity of your hand vs a two pair but I think to a 3bet shove you're only getting called by two pair, sets or straights and a very occasional flush draw that beats yours. I think betting and calling BB's raise being in position seems like the best play here