I actually don't know for sure if you'd be able to get the diff out or not, in situ, especially given that yours is a Remote type shifter, which means you have studs - long studs. You'd need to remove the manifolds too so you could remove the knuckle joints and get to the top stud nuts.
But after you have it out (assuming that could be done), you'd be able to clean it all up OK, but man, assembling it would be worse than making a pass at the mother in-law!
Pull the engine, then you'll get a good look at everything, and be able to clean it and see what you're assembling. It will be considerably quicker too.