You won't damage the 223 can shooting 22lr through it. What happens is buildup. And if you can remove that buildup, you can keep the 223 can in as-new condition. In order to remove the buildup, you will need to use one of two easy methods: ultrasonic or "the dip." The dip is a 50/50 solution of regular white vinegar and regular 3% hydrogen peroxide. It will dissolve almost all of the lead in the buildup (regardless of whether you are using it on a 223 can or a pile of dust on the floor). Beware of all this dissolved lead - it can soak through your skin and cause multiple health problems. Once you have "dipped" your can, you have to manually remove all the remaining buildup. Minor manual agitiation, such as a dental pick, will remove it. You can also place the item in the ultrasonic at this point. BEWARE that aluminum is not safe for use in the dip or an ultrasonic cleaner; these are STEEL ONLY operations.
So, if you have a steel can, even if it doesn't disassemble, you can all but fill it up with 22lr buildup, dip it then run through the ultrasonic and have a mostly new condition can. A good ultrasonic will cost you north of $500. That's about what a good 22 can costs. You make the decision.
My customers give their newby buddies their 223 cans for use on their 22s while they use their 22 cans. All ears are happy and the newby becomes my customer because nobody likes to use a 223 can on a 22 when a 22 can is available.