It took a bit of reading and experimentation to figure out how to do this, so summing it up here to help.
Decently describes how to save panel configurations, but what it doesn’t say is that until you save it to one of these panel configurations, there is no habpanel.config file to backup. Selecting the “Save the current configuration…” at the bottom of that page will create a file under “{habpanel_home}\userdata\config\org\openhab\ui” called “habpanel.config” that you can then backup.
This is also necessary to share your configuration with other devices, but backup was the main goal for me.