In theory, if you want to avoid mods, you can:
1. Use "save as" to create a new copy of your save. You don't want to mess your existing save if this fails. I'll call this "Save A".
2. On Save A, have the sim you want to change branches leave their job, join a different one, empty that sim's inventory, then export the household to the Library. Save the game and go back to the main menu.
3. Start a new game with the exported household, but only keep the sim you wish to change the career of. You should now be able to leave the bogus career you joined before and join the career you wanted to change the branch of, starting from the start. As soon as the sim is in the correct, non-branched career, export the single-sim household again.
4. Go back to Save A, delete from Manage Households the sim you changed the career of, and merge the version that is in your Library.
If everything went well you should have your sim in the desired career before the branch, with as few side-effects as possible — though the side-effects are likely to include the loss all the relationships of the sim that you messed with. The last time I tried this was 4 years ago, though, so I'm not sure it still works. Even if it works I would keep the old save for a while, in case you decide the side-effects weren't worth it.
Explanation: saving a household to the library only keeps part of the sims info, and the past careers weren't kept, at least last time I looked at it. The trouble with this method is all the other information that isn't kept.