I've just added support for /.well-known/button.json with a button of my own.

An image of an orange furry dragon on a purple gradient background, next to it is 'Dustys Corner' written in highly legible cursive, bouncing.

This site uses Jekyll, so I basically just created the file manually where it should be, and explicitly included .well-known in the config.

include:
  - .well-known/