A good theory, but it seems the official reason is just that it was a M.A.C.O. ship, and that's why the registry number system was different. See this article:

And if you think this is just our speculation, worry not: we got a note from Dylan Highsmith, one of the lead picture editors on STAR TREK BEYOND, about this very issue.

If you want the official explanation on the Franklin and it’s warp factor: it was a M.A.C.O. ship (or a United Earth Starfleet ship that housed M.A.C.O. personnel at times) that predates the NX-01.

When the UFP Starfleet is formed, M.A.C.O. was disbanded and the ship was reclassified as a Starfleet ship [with the USS identifier]. The ship is then “lost” in the early 2160’s.

It was important to everyone that the ship, like Edison, predate the Federation; that thematically, the ship mirrored an earlier time in history and served as a bridge in design between then and the NX-01.

Doug [Jung] and Simon [Pegg] may have worked up something [on an official launch date], but if they did it never made it to script or screen.

Either way it predates the NX-01, and was reclassified after the UFP is formed.