Well the "time-warp factor" was not the Spore Drive...two years too early.
USS Franklin could have easily been a competing engine design developed parallel to the Archer Warp Five drive. It even could have been the backup design. "Archer's drive is too ambitious reaching for warp five, so lets focus on getting to warp four first". It might use different principles to achieve the same general things...faster than light travel. It is entirely possible that while the Archer-Enterprise drive did work better, is wasn't able to handle larger starship like USS Kelvin. But the Franklin style drives could be enlarge to that scale. Thus Starfleet has some really large ships from time to time, while most ships use engines based on Andorian, Tellerite, or Archer drives, the huge ships use Franklin drives instead of the Vulcan ring drives. Than Nero arrives, and starships need to be scaled up a lot to counter the potential threat. Resulting in a lot of Kelvin-based starships and the next generation of that engine style is the new Constitution-class ships like USS Enterprise. Delayed in this universe because of the need to upscale the ships and come up with a new scaled up warp engine to power it. (The Spore Drive being put on the back burner as the new drives got ships their pretty quick compared to the earlier Archer based drives).
Like Enterprise (NX-01), the USS Franklin might not have reached warp four at first. Making it only after some improvements were made just prior to the launch of Enterprise, which could comfortably make Warp 4.5 from the day she launched, but had a really hard time breaking Warp 5, managing Warp 5.06 eventually, and after the Xindi mission, her first major engine overhaul could make Warp 5.2 for a little but, as could Columbia (NX-02) So it seems reasonable that USS Franklin, back in the day, could only cruise at Warp 3.5 or even Warp 3.8, with burst at Warp 4 or Warp 4.1 or whatever.
If that was the case (if she was MACO or Starfleet, it wouldn't matter), Enterprise could head out to the Expanse at Warp 5 for seven weeks ("The Expanse")...plus whatever time they spent going from Earth towards Vulcan first. Franklin would be lagging behind at upwards of Warp 3.8 or even warp 4. Enterprise, even though it is only "one warp factor" is twice as fast as Franklin. Franklin would not reach the Expanse for another seven weeks. Earth doesn't have time to wait. Enterprise discovered their first real clue six weeks after entering the Expanse ("The Xindi")...If Franklin was also coming along, she's still about a week away from the edge of that space.
