You can take your existing installation and copy it over to the new machine. The only thing you need to do then is copy this script:
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
Save it as "fs2.reg", and execute it. This will add a bunch of keys to the registry that FS2 expects to see. Once that's done, open the Launcher and have fun.