I'd try a clock shop for help. I seem to remember reading about it and the oils used are suppost to be very stable and not gum-up (obviously).
Another thing I read about and have used sucessfully (in other applications) is a "watch oiler". Take a small piece of wire (22 or 24 gauge works well) and flatten the end a bit. I did this by placing it on a vice and tapping it with a hammer. If you can get the end of the wire sort of diamond shaped, that's best. Put a tiny bit of oil on the end of the wire, then touch it to where you want to apply the oil. This works very well in controlling the application of minute amounts of fluid. I have used this trick for applying very, very small amounts of lubricants and adhesives. It works well.