With a central module hosting the game logic, the website could either use script tags or use a tool like browserify to include the game logic. If logic changes in the core repository, then the website and CLI applications would receive the same changes.
Sorry, I think this sounds great, but I really don't understand Node.js or how to do this, and I'm not comfortable working with things I don't understand. If you want to do this yourself, you could fork the project. I would love to see this done. Thanks