This is a cool idea. Do you have any suggestions for how you would wash sand? I can't really picture perforation fine enough that the sand wouldn't leak through.
I did it with pea gravel, which was what my dog was used to when I got him (he had lived in an outdoor pen). That worked well with 1/8" drilled holes.
Use a coarse grit sand will help, as it should be less likely to go into suspension in the water (muddy water effect).
If the cat will tolerate sufficiently coarse sand, you could use a piece of screen cloth on top of the perforated bin.
There is a beer homebrewing technique called brew in a bag, that could maybe be applied, and can be very cheap. The easiest way to visualize would be imagine a giant cotton trash bag, opened over a trash can. Inside is the sand, and also your soapy water. When you want to drain the water, you lift the bag up. Water flows through, the sand does not. Sewing up a bag is optional, you can just fold it to fit.
Maybe try a polyester sheer curtain, sometimes called voille. I got one at a department store (walmart, target, something) for about $10, and its lots of perfectly porous fabric. Polyester is nice in this case because it does not absorb water or rot. When brewing I just attach it to my stockpot with steel spring binder clips. The trick would be figuring out how to keep your cat from shredding it at the edges. Maybe a 3 bin system, where the top bin has the whole bottom cut out and only the sides remain to cover the filtering cloth. Another way would be to only use the cloth when washing the sand.
It is a setup that uses lots of water, to be sure.I cant see washing the sand for less than 10 gallons used, and it would be easy to blow through 40-100 if not careful. Perhaps the fish aquarium folks have some easier more efficient methods.