Thanks Toadies. I have a front-loader washer, so soaking it in there isn't an option, and I don't know how gentle it would be. I could soak it in the sink I suppose. It's not quite rigid, but it's stiff enough so that the mouth and inside stays open.
What I do now to wash it is wash it in the sink- I use handwashing liquid, lather it onto the fish, soak the fish for a few minutes before rinsing thoroughly (which takes a long time, it soaks up a lot of water) and squeezing it out a bit. Then if it's a nice day, I put it on the deck to flat-dry almost, leaving the mouth open. If not, it has to stay inside in the laundry room on a towel or paper towel (which gets changed when soaked). Either way takes a day or so to dry. Quite a bit of work for the one cozy.
I hadn't actually thought of the clothes line (which is how I dry the rest of my piggy laundry on nice days), I'll try that today when I plan to wash the fish.
Another question about it, how do people get the hair/poop out from the inside while washing? I find it still stays a bit dirty towards the back/tail of the fish, even with lots of rinsing near there.