When a backup starts, the little Performing backup window displays in the lower right corner.  When clicking on the Detail button in that window, this window disappears and the larger user interface appears.

Is there any way at that point to close the main interface and manually redisplay the little window on the lower right again?

Yes. You can simply click the OK button to close the backup progress window. Do not click Cancel, click OK, and the task will continue running in the background.