I think the simplest approach might be to add a menu item that saves the current document and then invokes a shell script with the path of the document. This would only need to touch a few lines of the C codebase. The shell script could then handle all the minutia of retrieving credentials and manipulating GitHub web APIs, and it would be easy to tweak the script to suit other deployment scenarios: upload to Neocities, rsync with a VPS, etc.