The screw is there to put pressure on a steel pin in the nut. It locks it into the bore. It doesn't necessarily have a through hole. your set screw should have a point on it, that pushes the pin.
Yes, it's a tiny set-screw in the center of the screw that retains the nut. I'd like to eliminate having to remove that screw.Is there not already a provision somewhere to oil the cross feed lead screw?
The only reservation I would have about a ball oiler is that dirt/crap might work down through it and get into the nut. It's in a pretty dirty location as that area gets a lot of chips, cutting oil, etc. built up on it while machining. That dirty cutting oil could work it's way into a ball oiler I'm afraid. YMMV.
Ted