I think you’re going to have a bit of trouble doing this from the ST platform, as you’ll have to set up port forwarding through your router if you want this to work. As @slagle mentioned, it might be more helpful to have something sit on your internal network as a server that can turn the xbox on for you.
If it helps, I just made this module for node, which can easily be called by using a simple Express server: https://github.com/arcreative/xbox-on
You would just need to create one route (preferably with some sort of authentication), then forward a port on your router. You can then hit this endpoint using a groovy script. I’m not familiar with HAM Bridge, but it seems like the same basic approach.