I am having a issue running this game in 1920x1080. I am using a 34" 1080i hdtv on hdmi as a secondary monitor and every time I set kotor to run at that resolution it starts the into movies fine and when I hit the menu it goes blank and the tv acts if it lost the connection, showing the input channel. I have run other games in the same resolution like fallout 3 so I am not sure what the problem is. I got the same issue in kotor 2 as well.
Sounds like an out of range problem. Perhaps someone more authoritative can advise.
I would recommend taking a screenshot of the movies and the menu screen using the Printscreen button and pasting into paint. This way you can confirm the output resolution.
KOTOR lets you select a resolution and refresh rate so that may be the problem. There is a .ini file (swkotor.ini) in the game directory that lists the resolution and refresh rate. Your refresh rate might be set too high for your display.