Since writing the above, I have realised that there is a much better way to resolve the problem.
Eraser does not show the popups if the Eraser program is open. So if you have, say, Photoshop open full screen, you can use the tray icon to open Eraser. You can still use the context menu to start Eraser tasks if you prefer. Once the task is started, you can click on the Photoshop window to make it sit on top of the Eraser window. The Eraser task will complete without popups.
Note that running the more disk intensive Eraser tasks (erasing free space or large amounts of data) is probably best not done when another program, particularly one as resource intensive as Photoshop, is running at the same time.