I could remove the fan and use the channel to outside.
that's the ticket. remove the cover for the fan to see if it exhausts to outside. you may have to remove the fan to be sure. if it does, just scrub the air and use your existing duct to exhaust.
you might even have a cage type fan. it's a long shot. looks like a big snail. it's an old design and can be noisy, but is the same design as standard grow space extraction fans. especially earlier ones. i would not use it, but it would make swapping in a proper one easy.
our washroom has a cage fan. but it's like listening to a freight train. the grow fans are way better.