According to Memory Alpha, the NX Project was an attempt at breaking Warp 2. The project was in 2143. So, as the Enterprise was the first Warp 5 ship in 2151, I can narrow the Franklin's launch between 2143 and 2151.
The problem with that is that the Franklin is referred to as the "U.S.S." Franklin, the prefix of which wasn't given to Starfleet vessels until after the formation of the Federation per ENT. Of course, this could just be a mistake on the movie's part.