you don't need to remove that box to take the coin door out of the cabinet...if you take all of the screws out that hold those silver tabs in place, the entire assembly should lift out
+1
As soon as I took out the screws and the silver tabs, the door and box pretty much fell out. You can see the lip carrying some of the weight of the box in the first photo.
You have to think, the coin door had to go in somehow, it also has to be repairable or replaceable in some fashion. So permanently riveting a door in place by sandwiching the door, wood panel and coin box probably isn't conducive to replacement and repair.