90's W/B, for the most part, had a white membrane applied first, then the artwork was screened on. I've redecaled several 90's W/B games and at first I removed the artwork and the white membrane by using a heat gun and putty knife. Worked fine, but taking a cabinet down to bare wood, you can the have the issue of raising the grain that's a bitch to sand out. Then it hit me, why not just sand off the artwork and leave the white membrane which it appears is where you're at. All you're looking for is a smooth surface and once you've filled in imperfections and sanded out the ridges in paint, from the silkscreening, you're there. Think of it like sanding a car to repaint, the last thing you want to do is to get through the primer to bare metal. Here is a Fish Tales I redecaled for reference.
http://timhanian.smugmug.com/Hobbies/Fish-Tales-W-NOS-playfield/12071033_S4RcXF#!i=857297966&k=DpcL2LV

Good luck with the resto.

Yep. A bit of fine sanding after and they come out smooth as glass. Oh and I've done several this way, over several years, with no problems.