Unfortunately, no, this sweepstakes is not for real. No legitimate contest requires an up front fee of any sort in ordert to collect winnings. When you are the real winner of a sweepstakes, the prize is already yours. Any transfer expenses get taken out of your money on the contest end, and you get all the rest of it when you show proof that you entered the contest.
Scammers always want you to believe that you can win a contest just by living, breathing, and having the right mobile phone number or email address, but you cannot. You must always enter a contest to win it. And you must always show proof of your entry to be able to claim your prize. Without that, they won't give it to you. That's how real contests prevent fraud.
If you did enter it, you would have your half of the entry form with your number information on it. If you don't have it you cannot win or claim it. Also, there is no such thing as an international sweepstakes or lottery. That's an immediate sign of a scam. Different countries' regulations make a legitimate contest a legal impossibility.
US Federal agencies/agents have nothing to do with the disbursement of contest awards and have absolutely nothing to do with the Caribbean, where they lack any jurisdiction at all. A US contest run out of Las Vegas would be reached in Las Vegas, not on some island outside of US borders.
The Better Business Bureau warns consumers of the Publisher's International fraud here.
Hang on to your money. This is a fraud and you will see no winnings. Report the scam to the FBI at IC3.gov and the Federal Trade Commission at FTC.gov. The only way you'll get back any money you spent is if the authorities can find these folks and bring them to justice.
I am very sorry to be the bearer of bad news.