It would be a significant amount of additional work for very marginal gains, but you could cut a small wedge out of each washer and modify your jig design so there is a small piece which sticks out in to the wedge cut-out to hold the washer in place. Almost certainly not worth the time, though. Your other option would be to buy square or other non-round washers, but again, you would obviously need to redesign your whole jig.

It sounds like your set up works fine with tape though. If you really mess up a washer by trying to punch it twice and being out of alignment, then you can just make a new one - it's not like you've messed up the entire back up as you might have done with a single steel plate.