Bikes aside, Banana Beach fits the bill perfectly for you, Susan - every single criteria you listed - amenities and locale.

Oh, and a great,(IMNSHO) staff who will take care of your every want & need . Fantastic brekkie options for two built into the room rates, which more than offsets any bike rental cost. Onsite travel shop who bends over backwards to help you. Great beach to sun yourself and right next door toone of the island's best swim spots. Two pools. Have not checked lately but I'm pretty darned sure the rates are equal to or better than most the other props listed. Easy walk up the beach to town with many options to stop & libate! ( No, I have no financial interest in BB, just my long-time experience)

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But then what do I know, I am but a mere caveman