... My one last point in desperation is whether there is any browser out there that could be persuaded to launch displaylessly on the server, open the address of the C2 'game' in order to trigger the battle using C2, saving the results each tick for players to later view? ...
Aren't you really asking for a server-side app that opens a web page whose source will be the index.html generated by C2 - that in turn fires up the exported C2 runtime, which will accept player input and display output from the server?
All browsers will do that, without any special persuasion - that's their job. The work is in building the server-side app. That's something that's really quite different from the runtime that C2 is designed to generate. I don't know of any C2-like toolkit that's designed to do that. Maybe one day someone will build one atop NodeJS ...