Unless you have welding equipment, I'm not sure I'd reccomend attaching from beneath, since a good smack to the button could cause it to fall into the CP. If you know someone with the right tools, you might be able to make it work. However, if you're just going to use glue or epoxy to hold the washer in place from beneath, it could potentially fall apart in the long run.
Basically, if you have a more thorough solution like welding, that'd be the way to go. However, given that it's a CP for a quiz game, it probably won't face the same kind of abuse as say, MVC2. Perhaps a strong epoxy and some bondo would be enough to get the job done, so I wouldn't write it off completely. In all likeliness, you won't see it fall into the CP in your lifetime, and probably not your children's lifetime either.