7 Mile beach is a public beach, but there are many access points. We stayed at the Comfort Suites and there is an entrance there, they have a pool (which you could probably use if you blended in with the guests, they don't check), and they have a pool bar/restaurant(great Conch chowder!) and the restrooms are right inside the backdoor of the hotel, so you could slip in there.
There are facilities somewhere on 7 Mile Beach, but I'm not sure where they are. The cruise lines have tours that go there, you pay admission.