Playing a big draw

Assume no reads.

Holdem No Limit - $0.5 BB

CO ($132.54) [Hero]
Button ($38)
SB ($32.39)
BB ($113.12)
UTG ($152.14)
UTG+1 ($173.71)
UTG+2 ($44.01)
MP+1 ($49.25)
MP+2 ($57.06)

Hero is CO with ,
Preflop: (9 players, $0.75)
3 folds, MP+1 calls $0.50 and raises $1, MP+2 folds, Hero calls $1.50, 2 folds, BB calls $1

Flop: , , :Kc: (3 players, $4.75)
BB checks, MP+1 bets $3, Hero calls $3 and raises $5, BB folds, MP+1 calls $5 and raises $15, Hero ???

I obviously raised for a freecard, but it didn't quite work out as planned. Now I *guess* shoving is the right move, villain has about 25$ behind, I have 12 outs to make a hand. Is that thinking correct?

Opp probably has AK at least.

You are behind, but you're nearly 50% to win. Pushing here is fine.

This would be interesting if his stack was the same as yours.

Given the effective stack sizes you really have no option but to push. You're only 50% to win if you see both cards.