According to wikipedia, St. Edmund's College is Green's sister college in Cambridge. I've no idea whether there is any reciprocal dining arrangement - how active the link is varies a lot between colleges.
edders' point relates to intra-Oxford partnerships though - St. Cross is a fairly small graduate college and it is quite common for them to have some sort of link to an undergraduate college (becomes important for sport, especially rowing) which might come with dining arrangements.
Green's main link is with Osler House, the clinical medical students' common room in the John Radcliffe.
The way to eat at another college is to have a friend there. To be honest, going to formal as a complete stranger to everyone there sounds like a nightmare anyway!