I believe that wool hats used to be steam-cleaned back in the old days. This was a common service performed by millners and/or dry cleaners, I think. I wouldn't try it myself, as it requires a hat form (kind of like a show last) to make sure the shape is maintained. I'd check around for any old-time hat dealers. Chicago is probably a good bet. Trying to contact a hat company might not be a bad idea, as well, like Stetson. One more thing, look around for theatrical or film production costume shops--there may be somebody who can help you out.
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Gemma Seymour (she/her) @gcvrsa