I have updated my card to Catalyst Control Center v.10.2. now and tried reinstalling the game with the updated drivers, but this time I don't even get to start it, it's just "Star Wars: Knights of the Old Republic: The Sith Lords encountered a problem and needed to be shut down". When the game scanned my hardware I got the following report, which contradicts the info that Catalyst Control Center gave me (Yeah, most of the info on Catalyst Control Center is in Swedish, but you can still understand where it contradicts Kotors report).
I'm thinking of getting a new copy of TSL, because I don't think it's my graphics card or the updated drivers that is the problem. I think the discs are corrupted, although they look fine to me (no scratches or anything else visible that could explain this). Also, I scouted around the net and found other people who had problems with TSL who also used Radeon X1950 Pro, but they only had small problems, like this one complaining about low resolution in the game: http://forums.obsidian.net/index.php?showtopic=51378&mode=threaded. If they can get Kotor 2 to run with Radeon X1950 Pro, then so should I.
Thanks for all the help anyway!