Hmmm, this one is a little more tricky compared to the FL-375 one, but not much.
As far as I can tell looking at the historic StreetView in each location, all but one of those FL-261 shields were added after a highway re-pavement/upgrades in that area. The only one that was there before 2016 (and that work), was the one going SB on US-319 just past US-90 (
Dec '07).
I'm honestly guessing that the newer shields that were posted could have been a contractor error. This isn't the first time in Florida that a random few shields for a hidden route get added after a reconstruction, and I bet it wouldn't be the last either.
Now, if the entire part of the Capital Circle that is FL-261 & US-319 got FL-261 shields, I could see adding it. As it currently stands, knowing it is meant to be a hidden number (almost all state routes under US highways are, unlike GA), and only a few shields were added, I'm not going to currently add it to the site.