Get a program called niftexture. It doesn't strip anything, but it does allow you to edit the paths. Simply strip off everything before 'Textures' and save.
I've used nifTexture for years, and it never once occurred to me it could be used that way.That's great, thanks!
I'd still like to figure out how to use the nif texture stripper though, because using niftexture to do it from now on is fine, but it'll take roughly forever to change all my pre-existing texture paths that way. According to the description, the texture stripper would apparently do an entire directory in a few seconds...