Yes, it is clear to me that the Moores were part rabbit.. there are so many of them. My Moores were in Pitt Co NC in the early 1700s. They migrated to GA, via SC, in the early 1800s. So, I think we're looking at two different lines here. Do let me know if your line was also in GA.
You might look at the Moore DNA project for hints. You might also see what the SC archives has in both their land files and will files. There was a Lazarus Moore that my Roden family had transactions with. Much better to look up a Lazarus than a John Moore.
There were a number of John Moores that served on both the Tory and Patriot side. Some were famous or infamous.
My first known John Moore died in Pitt Co NC in 1815. He is our brickwall at the moment. He did have a grandson named John Moore Sr.and he had a son named John Moore Jr. but that was in GA not KY.
Wish I could be of more help. All we "John Moore" researchers should start a club. best of luck, Cheryl Rhoden o7o