I would personally never rent a scooter in Bermuda, but since you've got a lot of experience visiting there you know the lay of the land. Last summer I think we walked past one of the places listed here
just a few minutes walk from the ship. I recall a woman calling out to us asking if we wanted to rent scooters.
Edit: Go here
and download the map. The place we walked past was #22 Eve's Cycle Rental, but the Oleander rental is just a little farther down the street at #19.