It would be much nicer if the backups ran as Local System in the background rather than as a foreground user mode process.

The UI to monitor an in progress backup in the background could be enabled having the dashboard/status window communicate with the background process.

That way, the background could run without the user being logged in, and the user could still see the progress window if they are logged in.

Currently, I have configured my backups to run in the background, but the consequence is that I lose the status window. And there's no way to monitor the progress and status of a background task from within the Reflect application either.

For now, this sacrifice is worth it, but it would be a much better architecture for the software to work the way I've described here.

I completely agree and would like to see this implemented.  As I understand it, an additional limitation of the current architecture is that if a backup job starts while a user is logged in, the user can't log off without cancelling the job, though I have not verified this myself.  Back when I was using Symantec Backup Exec (those were some dark times....), this limitation didn't exist because the backup engine ran as a service, the credentials for which could be configured rather than being forced to be Local System, and then the GUI application was a separate process.