Biggest problem will be not getting removed after 26 weeks (leaving for a day or so snd coming back in used to work - now with the computerised system they catch everyone trying it).
If you live in the US, think tax, after 26 weeks here you gecome liable to psy income tax in Barbados on worldwide income.
US$3,000, yes you would just about manage on that, but not if you want to live the expat lifestyle, you'd need $5,000 plus and not too much of the expat thing.
Internet here is excellent - think US75/mth
Think costs here for everything else US x 2 or 3
Rent, US750 - 1,000 for somewhere modest, and I do mean modest - up to 2,000 for furnished and fairly safe, I favour top floor apartments. A saving would be to find someone looking for a roommate - good to start.
There are hefty deposits on phones, electrical connections - off the bus routes there is the costs of a car - thing year one costs for a ten year plus clunker US10-15K
Your biggest problem will likely be the 26 week thing - there should be no work permit issues unless your work is connected to economic av0ctivity in Barbados.