I use a Logitech Extreme 3D Pro joystick. It works perfectly fine with vanilla, unmodded Freespace 2. But as soon as I install Freespace Open / SCP, I can no longer use the joystick in the game, it doesn't register at all. Any suggestions? Again, when launching through the "vanilla" exe, no problem, but when launching through the FSO exe, it doesn't work.

Have you tried binding the joystick assignments in the controls menu?
Joystickcontrol use to work for me in FSO, though it's been a few years since I last played it.

As a sidenote, I never liked joystick control in FSO.
Maybe they changed controlbehaviour in favour of mousecontrol for which FSO seems better optimized.

If you use knossos check in the launcher settings whether it's the default peripheral.

Hey mate, I just figured something out with my setup and maybe it will help.

When I launch retail FS2 and go into the setup (C:\GOG Games\Freespace 2 or wherever you installed it) and launch FreeSpace2.exe then click setup then click the joystick tab, it'll have a name like Joystick 1, make a note of that.

Launch Knossos then click the cog icon then click on Joystick dropdown and set you're joystick to the matching Joy-Number, for example, in the retail setup it says "Joystick 1" (my joystick is actually a thrustmaster blah blah but its not releavent) so I set it to "Joy-1" in the Knossos Joystick setup, make sure to save your settings.

That got it working for me, its strange because when I go to the options ingame it says "Joy-0" but its running on "Joy-1".

Hope that helps and gets you going.