Hi Matt,
Sure you can weld Stainless to copper or brass. Having said that, I would not suggest it be done in a load bearing situation. In the construction of argon backups, heat sinks and fixturing I have many times welded stainless to copper with TIG, using stainless wire or if heat imput needs to be minimized due to the small size of the part we sometimes even use High Temp silver brazing wire with TIG (no flux). If your more adept at silverbrazing you can join stainless and copper that way too. Remember this is not a recomendation to weld structural or load bearing items of dissimiliar alloys without alot more research.
Tell us how it goes or if you have further questions