Okay, I thought it was a fluke at first but now it's a recurring issue. I am using 1.5.24.0.5 (non-steam) version of Skyrim and helifax's fix for said version. When I first start up the computer and start the game with the fix it works just as described without issue. BUT, when I close the game & the fix and start them up again later I get issues. The HUD is on the far right monitor as soon as the game loads, it will go onto the center display if I enter and exit the map, but it's still pushed to the right by 115 pixels. I've tried manual settings, but it just keeps doing this until I restart the computer. At first, I didn't think the fix was terminating correctly, but I made sure the process was not in task manager before starting it again and then starting Skyrim which still gives me the issue. By the way, the fix does not conflict with a RivaTuner overlay when it is functioning normally, it just moves it to the left edge of the center monitor instead of the left edge of the left monitor where it usually is. So I've kept my RivaTuner overlay in the screenshots to show the offset better as it SHOULD be close to the left edge of the center or left monitor at any given time when the fix is behaving normally.
When Skyrim is first started:
After map is opened and closed:
It's like everything is behaving as if normal, but it's just shoved over to the right by one monitor. The issue where you say HUD winds up on the left monitor when exiting the map in this case is putting it in the center instead because it's off to the right by one monitor.
EDIT: Okay, I figured it out. The fix is not starting fast enough. I set the delay to 5 seconds and it stopped doing it. Still posting this in case anyone else encounters this.