SB ($31.88)
Hero (BB) ($28.84)
UTG ($12.57)
MP ($25)
CO ($31.73)
Button ($45.36)

Preflop: Hero is BB with
3 folds, Button bets $0.75, 1 fold, Hero raises $2.50, Button calls $2

Flop: ($5.60) (2 players)
Hero bets $3.75, Button calls $3.75

Turn: ($13.10) (2 players)

Hero...

should I give up on the turn?

I wouldnt bet again, when he calls the flop his most likely hand is an A or K, with the 2nd king on the turn he will probably not fold. I would check/fold

I would have called the turn given size of the pot

@spekkie: how can I call the turn, I have to either check or bet

@ Japmaster: he could have a flushdraw that he might fold if I bet

He could have a FD, if he has Q/J high FD he`ll probably check behind on the turn, if the diamond comes you`ll lose a bet on the river. But if you bet the turn you`ll lose 1 bet most of the time also because his most likely hand is an A or K. Therefore I think you`re better of checking.