1) using relative paths in the update-center.json allows to locally mirror the jenkins update site so that a jenkins installation that is behind a corporate firewall can still be updated comfortably.

2) also a file named .json should contain json - not jsonp.
jsonp should be served with the content type application/javascript and named .js

thus in case the requested change can't be made without breaking older jenkins version I suggest to add a new json file for use with newer jenkins version that could also fix part 2.