If you happen to have the access token (and, if needed, the token secret) handy, you can always create the templates yourself. The only thing you miss out on is the automatic handling of the connection and creation of the API binding. You could also use the UsersConnectionRepository to give you a ConnectionRepository and from that fetch the API binding you need. It's just more work for you that way.
I can appreciate the desire to be stateless, but a request-scoped bean doesn't necessarily imply state. In fact, that's one benefit of being request-scoped instead of session-scoped, is that you get a per-user bean that doesn't survive between requests. It's not much different than if you had created that connection repository bean yourself on each request and then used it to create the API binding. The only real difference is that you didn't have to create it...Spring created it for you. Yes, it gets carried around in the request, but it only lives in the current request, not between requests. Therefore, it doesn't break the stateless approach you desire. Of course, it will create that bean on every request, even if that request doesn't need it. Maybe that's what you're striving for.
Spring Social Project Lead