Would double sided carpet tape work?

After you drilled your holes, you could peel the tape off of both the vinyl and the steel washers.

Double sided carpet tape has a good initial grab, and it's meant to be pulled off the floor and off the carpet backing without leaving any adhesive residue behind.

There is also double sided mounting tape for mounting mirror tiles on walls (or ceilings). Mounting tape is much less expensive because it's a smaller roll, more difficult to remove and leaves an adhesive residue behind. Mineral spirits (often called "paint thinner") will remove the glue residue from double sided mounting tape.

A third option might be to use acrylic or silicone caulk as a weak adhesive. Acetone will remove acrylic caulk and shouldn't harm PVC (vinyl) but I don't know about polycarbonate (check first). Acetone is commonly sold as nail polish remover, so you can buy a small bottle of it for a few dollars, or a gallon for about $20. Silicone caulk can be removed with mineral spirits. Dow Corning's "Silicone-Be-Gone" is nothing more than gelled mineral spirits which shouldn't harm either vinyl or polycarbonate plastic (but check first).