I'm going to suggest a different than normal solution to this problem.
Use this as a team code event. Have everyone check-in their code who can, then help others who are still working with the file. Once everyone relevant has their code checked in, find a conference room with a projector and work together to start moving things around and into new files.
You may want to set a specific amount of time to this, so that it doesn't end up being a week worth of arguments with no end in sight. Instead, this might even be a weekly 1-2 hour event until you all get things looking how it needs to be. Maybe you only need 1-2 hours to refactor the file. You won't know until you try, likely.
This has the benefit of everyone being on the same page (no pun intended) with the refactoring, but it can also help you avoid mistakes as well as get input from others about possible method groupings to maintain, if necessary.
Doing it this way can be considered to have a built-in code review, if you do that sort of thing. This allows the appropriate amount of devs to sign off on your code as soon as you get it checked in and ready for their review. You might still want them to check the code for anything you missed, but it goes a long ways to making sure the review process is shorter.
This may not work in all situations, teams, or companies, as the work isn't distributed in a way that makes this happen easily. It can also be (incorrectly) construed as a misuse of dev time. This group code needs buy-in from the manager as well as the refactor itself.
To help sell this idea to your manager, mention the code review bit as well as everyone knowing where thing are from the beginning. Preventing devs from losing time searching a host of new files can be worthwhile to avoid. Also, preventing devs from getting POed about where things ended up or "completely missing" is usually a good thing. (The fewer the meltdowns the better, IMO.)
Once you get one file refactored this way, you may be able to more easily get approval for more refactors, if it was successful and useful.
However you decide to do your refactor, good luck!