Re: Oiling Fretboard - remove all strings, some strings, or no strings?
I would suggest you down tune a bit, remove the springs in the back, and pop the whole tremolo out, strings still attached. Polish up the frets and oil the board and then pop the tremolo back in, reattach the springs and bring it back up to tune. Then you can change the strings one at a time. I personally think it would take less time to do this than it does to get it balanced back out after removing all the strings.
Edit: I just realized you said they are both fixed bridge. No need to worry about the trem in that case.