I can only repeat and confirm that if you don't see the registration window, it's because you have another FSX Add-on that is trying to force itself on the foreground, so the registration dialog is behind the FSX screen.
It's not something you have on the desktop, it's surely an FSX add-on, and programs known to cause this are online flying client with their chat window that tries to be always on top (a very questionable behavior).
You can try to bring the registration window back on top, by pressing CTRL+TAB or ALT+TAB to cycle between opened windows, immediately after you press the "Register Serial" button.