Update: Well, I went back to the tree recently; and the property management definitely is gearing up for the tourist season. The brush was trimmed away; the tree was pruned, and sadly, all of the figs were gone. I don't think they could have all ripened this early, but every one of them was missing. Nothing on the ground either. Very disappointing.
So, I guess my mystery fig will remain a mystery until I get it to fruit. Glad I grabbed the sucker and some green cuttings while I could. So far, they appear to be rooting OK. I'll go back later in the year to get dormant cuttings to share.
For now, I'm calling it "Kure Beach Unknown," for the area where the parent plant is located. Kure is pronounced like "curry." I've also seen a number of other trees along the coast, so I've got other ones to sample later on.