Okay, for now, until you can work out a solution, you can always try putting the wet clothes on drying racks - plastic hangers will do - and leaving them someplace warm in the house.

It sounds like you need a way to completely encase the dryer that's kitty-proof. Plywood and a few nails? Know anyone that is "handy" that might have some scrap lumber around? If not, maybe call the high school, and see if the woodshop kids would be interested in a project in exchange for a batch of cookies and some class credit ... You could send a note with a "Save this Kitten's Life" and a picture of him looking his most adorable ....