Mom always put dad's caps on a small sauce pan to dry so they would hold their shape. I have done the same thing in the past.well using the automotive interior cleaner did the job..!!
spray on, brush it in, wipe it off with a towel...just like i would do to clean the carpets or seats.
i then placed the cap on the room heating radiator, to dry, and no issues with the bill getting all out of shape, or the cap losing its shape.
i still will order one of those washing devices