Have you tried taking it to the dry cleaners? I’ve had some success in the past with textile arts using the solution inside the Dryel dry cleaning kit; it required plenty of blotting and patience. FYI: If you go to the cleaners, they will ask what made the stain (e.g., oil, dust, ink, etc.) and some knowledgeable ones will recommend how to clean it or refer you to someone who does.
You’re correct about the backing (hardboard) in that it must not get wet – I could be wrong but it looks like MDF (medium density fiberboard). They will swell up upon contact with water. Good luck.
EDIT: Just wanted to say that you have a lovely piece of folkart.