I didn't know how to plan the hand at flop.

Opponents are fishes with VPIP over 60%.

I decided to lead, wanted button to fold and after that I was planning to go slowly AI with small blind with tiny turn and river bets. Against button I was too deep to go happily all-in.

How should this be played?

I think it plays alright as either a bet or a check. I wouldn't mind a smaller sizing either, as this is a lock down board and villain has maximum seven outs, and if we could encourage someone to draw to a straight that would be delicious. Really this is a WAWB situation and a very static board. Especially 723 you have to think the pair up draw is less likely than normal given that the pot was raised, and even loose guys do fold some hands. With this hand in specific, and with your range in general, this flop texture (monochrome) favours small bets.

Depending on your opponents and how they play, you can be checking to call or checking to fold on the flop.

Don't like it. The board is such that it allows for no equity shifts (so "lockdown"?) - whoever is ahead is basically 100% ahead always

While our hand is usually the best here, it isn't strong enough to go for stacks. It's a decently high SPR situation three ways: J high flush ain't gonna be the hand to stack 'tards with by value-barrelling three streets..

Basically I think your line here should be to check and start praying if btn begins to bet multiple streets. If it checks around and sb starts betting turn, get ready to call down. If it checks around on flop and sb checks turn you can probably valuebet half-pot-ish twice.

Basically: Proceed with ****ing caution m8...

You could squeeze preflop if you want btw, but sb stack size probably makes him leech EV off of you and btn in that spot