I suspect that from brew to brew the effect of contact with a zinc plated washer will be negligible. The zinc would (very, very) slowly dissolve under the effect of acid conditions and, generally speaking, zinc ions in solution are "not optimal" for yeast metabolism. But IMHO the amounts of zinc compared to the amount of yeast would again render the problem negligible.I was wondering what electrolytic effect a zinc/iron dipole might have, but again I don't suppose that would be significant of the course of 10 days or so. If the zinc washer was the only viable solution in the circumstances then I would certainly give it a go rather than not brew at all.
Would it be possible to coat the zinc washer in (say) epoxy resin? Would it be possible to make up a suitable washer by cutting a drilling some stainless sheet? It is certainly possible to obtain stainless steel washers of various sizes online.