You should choose a projection based on: 1.) your study area location, and 2.) what you are interested in measuring (distances, areas, or angles). Projections either preserve distance, area, or angle (or compromise to lessen all three distortions, but fully preserve none), so choose a projection that preserves what you are interested in, in your case distance. The smaller your study area is, the less dramatic the effect of distortion, so in your case you may never notice the effect of choosing an inappropriate projection. The farther your study area is from the projection's "sweet-spot," the more dramatic the effect on your measurements will be, so choose a projection suitable for Bimini, like UTM zone 17 as Robert suggests.
As for units, you can directly calculate miles from meters (meters * 0.000621371). You were seeing widely incorrect numbers using WGS84 for distances because it uses degrees.
If you're interested, you can go here to see the effect of trying to measure distances on an unprojected surface. All circles are 500km in radius.