Hey so I've just realised how much I dislike the option I chose. Is there a way to revert your save (Including your items and possibly companion affinity) back to when you had a choice to join the Institute, Railroad or the Brotherhood of Steel?
You are describing the reason why I never overwrite saves in games like fallout.
Treat saves in fallout like a checkpoint, you can't fix what you've done previously after you load a save, but if you have three saves from different points in your game you can load an earlier one from before you made the decision you regret.
This comes in handy in any game with branches or real time cause-and-effect: fallout, any visual novel, most tycoon games, etc. Save often, and don't overwrite, so if something goes wrong you can go back, and if there's a branch or a different approach you want to take, load a save from before you committed and you'll be able to see all consequences or endings without having to replay the earlier content.
I am fairly positive the only way to do this would be to go back to an auto save or manual save from before you chose.