I don't clean my own filter any more, but back when I did, I would do it on the
lawn, and it never seemed to adversely effect anything. Its so fine -- I don't think it would hurt much in your compost.
Now we've got a salt-water pool, so I'm a bit more careful about the salt on things. (As if splashing chlorine around was any good). I love how much better the salt system is --- easy on your eyes and the kid's swimming suits, etc., but I wonder about the long-term effects of salty pool
water tracking around and that salt accumulating and washing off the
concrete into the plants beside the pool over the years. We shall see. Hopefully, the fungal network will take care of it.
So if I had a chlorine pool, I'd wash out the filter over the compost pile. Give it a shot -- see what happens. But not for a salt-water pool. I wouldn't want that salt accumulating in my compost.
"The rule of no realm is mine. But all worthy things that are in peril as the world now stands, these are my care. And for my part, I shall not wholly fail in my task if anything that passes through this night can still grow fairer or bear fruit and flower again in days to come. For I too am a steward. Did you not know?" Gandolf