I also have the Iwata, and have been using my ultrasonic tub to clean it for a long time. I just use a jewelry cleaner powder of some sort for ultrasonic cleaners. When I return home, if I think of it, I'll advise the name of the product.
This may be obvious, but I disassemble the a/b before putting the components in the tray. And yes, I put the stripped a/b body in too.
I have no idea if this procedure is correct, and I am no airbrush expert (we have a couple of them here, but I'm not them), it's just what I've been doing for over 10 yrs, and I'm still using the same Iwata, and it still works as new.