Okay, here's how I fixed this.
Step 1. Run the fs2.reg file from the steam version. If you do not have it, copy the following text into notepad and save it as fs2.reg, then doubleclick on it.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Volition]
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Volition\FreeSpace2]
"EregCount"=dword:00000003
"Videocard"=hex(2):44,00,69,00,72,00,65,00,63,00,74,00,20,00,33,00,44,00,20,00,\
2d,00,20,00,50,00,72,00,69,00,6d,00,61,00,72,00,79,00,20,00,44,00,69,00,73,\
00,70,00,6c,00,61,00,79,00,20,00,44,00,72,00,69,00,76,00,65,00,72,00,20,00,\
28,00,33,00,32,00,20,00,62,00,69,00,74,00,29,00,20,00,28,00,31,00,30,00,32,\
00,34,00,78,00,37,00,36,00,38,00,29,00,00,00
"CmdLineOptions"=hex(2):00,00
"Interlaced Movies"=dword:00000000
"Movie Fast Clear"=dword:00000000
"D3DFast"=dword:00000000
"D3DTextureOrigin"=dword:00000001
"ForceWFOG"=dword:00000000
"Detect Glide"=dword:00000000
"Detect D3D"=dword:00000001
"DisableZbias"=dword:00000000
"Soundcard"=hex(2):44,00,69,00,72,00,65,00,63,00,74,00,53,00,6f,00,75,00,6e,00,\
64,00,00,00
"LauncherSoundEnabled"=dword:00000001
"EnableJoystickFF"=dword:00000000
"EnableHitEffect"=dword:00000001
"ComputerSpeed"=dword:00000002
"NetworkConnection"=hex(2):4c,00,41,00,4e,00,00,00
"ConnectionSpeed"=hex(2):46,00,61,00,73,00,74,00,00,00
"SkipVerify"=dword:00000000
"PXOBanners"=dword:00000001
"ForcePort"=dword:00000000
"Language"=hex(2):45,00,6e,00,67,00,6c,00,69,00,73,00,68,00,00,00
"StraightToSetup"=dword:00000000
"Gamma"="1.80"
"LastPlayer"=""
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Volition\FreeSpace2\Version]
"Major"=dword:00000001
"Minor"=dword:00000014
"Build"=dword:00000000
Step 2. Open regedit (Windows-R, enter "regedit", press return)
Go to HKEY_CURRENT_USER\SOFTWARE\Classes\VirtualStore\MACHINE\SOFTWARE\Wow6432Node\
You should have a key named "Volition" there.
Step 3. Right-click on it, and select "Permissions".
When I did this, the "Group or user names" field was blank. If it is, click "Add".
In the dialogue that now pops up, Enter "ALL APPLICATION PACKAGES" into the "Enter the object names" field.
Click OK. The dialogue should now be populated.
Step 4. Go to the FreeSpace2 subkey under the Volition entry. Check permissions as above, and if necessary, fix as above.