Looking through the XirSys API, I don't see a way to simply get my IP address as it visible from XirSys.
Is that possible to get via the exposed API?
The XirSys STUN servers do exactly that, but it's not something you can request from an HTTP request. If you're using a middleware server (for security), however, then you can do this in pretty much any language. See for example: How to get the client IP address in PHP?