I wouldn't recommend shooting too much .22 LR through a .223 can. Most .223 cans aren't designed to be dissassembled; the pressure is high enough most of the carbon and fouling blows right out of the can, and only a relatively thin layer ever accumulates inside.
But when you shoot .22 through it, the pressure isn't high enough and most of the fouling stays inside the can. And it's made worse by the fact that the .22 is a dirty cartridge and leaves a lot of lead fouling.
So while you normally don't need to clean a .223 can, you will if you shoot .22 through it, but it will be difficult to get the lead fouling out if it can't be dissassembled. Besides, .22 is usually quieter when shot through a can specifically designed for it.
0331: "Accuracy by volume."