I'm trying to start the game via the Morrowind Script Extender, which was installed as instructed and which worked in the past just fine. No problems whatsoever. Recently it's stopped working, displaying the error message, "Could not start Morrowind: Could not start Process: The directory name is invalid." I haven't altered the Morrowind files in any way since it was working last, in mods or file location.

I've no idea what it's up to, but I'd like it if it stopped seeing as I have a mod to test. Any thoughts on what to do?

I think you should stick to the Morrowind Launcher and avoid MWSE, because if you're using the http://www.gamesas.com/index.php?/topic/1113265-morrowind-graphics-extender-mge/ which has an internal MSWE built in.
If I where you I would install the MGE right away and *almost* forget about the MWSE apps and play Morrowind as it was intended.