The real question at this point is how you would get a wire out there. I would seriously consider just going on the side of the house, near the rear or in the outside wall of the screen enclosure kick panel. A lot of time I see EMT strapped to these enclosures for things like this.
In fact I am using EMT for the overhead lights in my pool cage but it is 14' tall. My outside receptacles are on the corners, attached right to the cage. I was able to get the raceways underground before the cage was built for these so it is cleaner than strapping EMT to the wall. At my FIL's house we ran them right along the joint at the base so it isn't totally ugly.
