Very nice. I've never seen one either.
My gut feel is that this may not have started life as a button, but has been converted from a medalet, non-currency token/counter or something of that kind.
Note that 26th June 1830 was William's accession date (probably expressed on the button as an abbreviation for the less-used term these day: 'ascension') and not his coronation date. We have a "the King is dead, long live the King" protocol (as happened after Elizabeth's recent demise) but the actual coronation happens on an agreed date after that... and often several months later. Charles' coronation for example might not be until 2nd June next year.
Coin-like items of this kind were widely produced as souvenir memorabilia for coronations, Royal weddings, anniversaries etc and occasionally for the initial accession too. At that time there wouldn't yet have been any coinage portrait to copy, which might explain why William isn't depicted.