I don't want to destroy BoS, Railroad, or Minutemen. Is there a way to do that?

Currently my quest lines are:

BoS: Liberty Reprimed

Railroad: Underground Undercover and Variable Removal: University Point

Institute: Battle of Bunker Hill

Minutemen: None (I've already retaken the castle and have over a dozen MM settlements.)

Note: My favorite companions are Cait, Piper, Danse, and Dogmeat. I don't want any of them mad at me.

Note 2: I ratted out The Institute's plans to both BoS and Railroad already.

Yes, it's possible to get an ending like that. I actually did a write up of how I got it here - http://www.gamesas.com/topic/1587288-3-faction-ending-with-max-of-faction-quests-and-all-perks/

Far as I know, only the Minutemen allow you to destroy the institute while leaving the other factions in tact.

You're at a pretty good point, actually. If you side against the Institute in the Battle For Bunker Hill, you'll get a confrontation with Father where you can cut ties with him - this'll start the Minutemen path to the final quest (which starts whenever you're officially enemies with the Institute, so killing a named NPC will work too and you'll get other chances in the questline to go against them).

I'd suggest avoiding that for now until you complete Liberty Reprimed and Blind Betrayal for the Brotherhood of Steel, since those are the last in that questline before you're asked to destroy a faction you don't want to. You can stop now, but Blind Betrayal is a great quest that I strongly suggest you check out.

Thank you so much for the help!

Well, how you do in Battle of Bunker Hill doesn't matter at all. You can kill the courser or send synths back to The Institute. You can actually kill everyone (except the settlers and caravan folks present) or you can just watch and see if some drop legendary weapons. You can at any stage do "banish from the Institute", misc you get if you attack any at The Institute.

Cheers,

-Klevs