As a ham radio operator, I would agree that a repeater would be fun and useful. But there are a couple of drawbacks/limitations (in my opinion) that would restrict the installation of such a repeater.
1 - Electric power, there is very little of it. I believe everything on Loggerhead Key is powered by solar panels. I can only assume that the park management would keep tight controls on the loads to those panels.
2 - Height. The lighthouse is only about 150 feet in height. Not tall enough to reach Key West without some good engineering. They would need a directional beam pointed towards Key West. Also would need to have a high site in Key West to receive and rebroadcast. I think this is how they set up their UHF phone link.
They do have sat phones these days but cost is still a factor. Also, I understand that their VHF repeater is linked via RF to the dispatch center at Everglades National Park. I was told that they link it via the tower at the Flamingo site. It would be nice it you could not only confirm the use of the frequencies I have as being used for repeater use, but also if any of those are used as an RF link back to Flamingo.