“What would a couple layers of such mesh do offset slightly?”
They would be additive - but only if you had, and I’m estimating here, them nested with a foot or so in between layers. This might require 3 layers to get to something approaching mil-spec across a broad frequency.
If you are genuinely concerned - and if you want cheap, go with a steel garbage can - put raw steel wool fragments all around the lip of the can. Fasten down tight with a rubber-mallet. Only open when you need to....., keep in a dry place or seal the lip with caulk or other moisture barrier.
Arizona is pretty naturally dry.
However the garbage can lid etc. would not work for a solar array. Is there no way to protect an array out in the sun and operating from EMP?