Raise the river with King high. Odds to do so?

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Soft 10/20 game at the casino.

I'm on the button with K[img]/images/graemlins/club.gif[/img]-10[img]/images/graemlins/spade.gif[/img].
Folded to me. I raise. SB and BB call (both semi-loose, non-aggressive players).
Flop (6 SB) -> A[img]/images/graemlins/heart.gif[/img]-8[img]/images/graemlins/club.gif[/img]-3[img]/images/graemlins/heart.gif[/img]
Checked to me, I bet, SB calls, BB calls.
Turn (4.5 BB) -> (4[img]/images/graemlins/spade.gif[/img])-A[img]/images/graemlins/heart.gif[/img]-8[img]/images/graemlins/club.gif[/img]-3[img]/images/graemlins/heart.gif[/img]
Checked to me, I bet, SB calls, BB folds.
River (6.5 BB) -> (10[img]/images/graemlins/spade.gif[/img])-4[img]/images/graemlins/spade.gif[/img]-A[img]/images/graemlins/heart.gif[/img]-8[img]/images/graemlins/club.gif[/img]-3[img]/images/graemlins/heart.gif[/img]
SB bets. I raise...

Correct play on my part?
My thought here is this player either is determined to call down with a pair or is on the flush draw. The stop'n'go bet on the river, now pushes me in favor of believing he has just missed his [img]/images/graemlins/heart.gif[/img] draw.

Unforunately he may have picked up a smaller pair along the way. With only king high, would I be correct in thinking raising or folding (not calling) is my only option here? If he has no pair and was on a flush draw he has to fold. This raise may also perhaps make him muck a medium pair.

With the pot laying me 3.75:1 on my raise, I only have to make this player muck 1 in 3 times in the long run, to make this play profitable. The way the BB played this hand, what percent of the time do you think he would actually have been on a flush draw here? (I estimate 50%, but this figure is up for debate).

Thoughts on the whole hand?