Heeeyyyy... That's a good point!
I could glue the thin sheet copper to, say, a piece of hardboard (I believe it's also called masonite), and screw that into any inexpensive case. I seem to remember a link to making rack cases using metal two-by-fours or something like that. You'd have an inexpensive and very attractive case.
Of course, the patina chemicals can be quite costly, but you don't need to use much of them.