Srry no HH at work and this hand is bothering me.

My image is LAGG PF not so much post-flop, I've been opening a wide range but basically been playing passively with weak-made hands. BTN is a TAGG who just lost a pot about 3 hands ago. BB is a total callingstation fish, he's the reasons I'm sitting, I've already stacked him when he two-barrelled 88 into a King High flop and I just c/c twice with AK in a 3bet pot.

Hero ($220)
BTN ($85)
BB ($70)

Fold to me in the CO with :Ad: :Ah:

Hero raises to $4, BTN raises to $12, BB calls $12, Hero raises to $28...

What do ya think? Should I just shove over? Flat-call? Make a bigger 4bet? What reasoning do we have for 4bet shoving?