Any ordinary taxi from the line waiting just outside of the doors of the Arrival Hall (local taxi drivers know the airline schedules better than the air traffic controllers) at the airport will be just fine. If you arrange for a "..private transfer.." through Pompano Beach Club (or almost any Bermuda guest accommodation...and probably cost more than the normal taxi fare)...all they will do is pre-arrange for an ordinary taxi to pick you up at the airport. The average taxi fare between the airport and Pompano Beach Club will run $40-45.
Years ago we used a company called LONDON TAXIS (now sadly out of business) that operated three(3--black, white & blue) renovated, 'boxy', Austin, '..London..' taxis on Bermuda. We used them two or three times for airport pick-ups (for the 'hoot'). At the time the largest automobiles on Bermuda were the Governour's Damlier-Benz, the Premier's BMW-700 and the local mortician's Ford Counrtry Squire station wagon...and these three Austin taxis. Sadly, the company is no longer in business but at least one of the old Austins (the white one) is still around in private hands and is occasionally seen transporting brides & groom at the head of local wedding processions.