Some more thoughts on sources of "shim" material.
One might even consider jewelry supply houses for copper and silver
of varying thickness and dimensions. Both are used in jewelry and likely come in a wide variety of geometries (rolls, rods, foil, thicker "plates", beads)
And, electronic supply houses and auto repair shops.
For example, digikey.com
sells copper tape in multiple thicknesses, widths, and lengths.
They offer a 0.004" x 1" by 36" copper tape for $4 (currently on backorder)
There's also 3M3190E-ND, which is about 0.004" x 10" x 10" square of copper for $5 (in stock)
They also offer tapes in actual silver (not just silver colored). Of course it's going to be a bit pricey.
If it weighs less than 8oz, then digikey can ship it via 1st class mail for about $2.41
You'd likely have to dissolve off the adhesive on these particular items, but this is just an example of what's out there. There are probably some that have no adhesive.
Or, you can get "EMI Gaskets & Grounding Pads" or "Suppressor Sheets" from a place like mouser.com
Dimensions, materials, etc. vary all over the place but copper "foil" type stuff looks like it's in the same price range as digikey.com
, but it comes in thicker sizes if desired, and...it also comes in squares, rectangles, etc. instead of a roll of tape...more like a thin copper panels it would seem.
Not as likely, but one might also be able to beg scraps from any sheet metal fabricators in your area.