Do the fabrics bleed? You can test for that with a wet Q-tip. If they don't bleed, no need for a Color Catcher. If they do, I'd wash them by hand with a Catcher.
Just doesn't seem like there'd be enough room for things to move around in a mesh bag and if there's a runner in there, it will sit on other fabrics no matter what you do. Just my opinion.
ETA: If one of your fabrics is a runner, you'll have to know which one in order to wash it over and over until it stops. Otherwise, it will continue to run when chewed.