I can't comment on whether there is a boat across the gulf, but I'd suggest that it would be quicker and cheaper to head by road to Bangkok and fly down the Phuket .
Another way of 'thinking outside the square', is use AirAsia to get to Phuket via Kuala Lumpur. This would involve two flights and depending on when you flew, cost you somewhere in the region of US$90-100.
If you explore this option, remember to leave about 3 hours between flights at KL.
well there's no shortage of Cambodian fishering boats plying Thai waters ;-)
Save that, as per above, overland to Bangers and fly from there. Road to Poipet is now good and pretty, the schedule gods smile upon you and you'll be in Bkk for a late lunch.