I've found a utility called Flawless Widescreen that automatically makes KotOR work at 1366x768 (and presumably any widescreen resolution). (@Q: Is this the utility you mentioned before?)
Here's a description copy/pasted from a post I made in another thread:
This program is awesome. It's made for making games run properly at "super-widescreen resolutions", but It works for 1366x768 as well (and presumable any other widescreen resolution). It has a specially made plugin for KotOR, that automatically makes the game run at your desired resolution, while dynamically creating properly scaled menu backgrounds and HUD item placements. Movies are scaled properly, and most impressively, *dialogue is displayed correctly*. It does all this through memory manipulation, so there's no need to patch the game's exe like with UniWS (this also means that it works fine with the Steam version of the game, i.e. no need for cracks). You don't even need to do any swkotor.ini editing - FWS does it all for you.
So here's how to do it:
tl;dr: Install Flawless Widescreen + KotOR plugin; run KotOR with FWS in the background; change resolution in-game to 1366x768; play.
1. Obtain Flawless Widescreen
2. Extract the archive to its own folder
3. Run the program (confirm Windows UAC)
4. You will be greeted with a splash screen asking if you want to change any settings. You don't need to do this, so click "No" (or click "Yes" if you want to see the settings. But don't change anything in there)
5. In the list found on the left side of the program window, find the entry "Star Wars: KOTOR"; click on it
6. Flawless Widescreen will now download and activate its KotOR plugin.
7. Now, on the information pane on the right, you'll see that FWS requires you to have run the game at least once before using FWS. Make sure you've done that.
8. With FWS running, run KotOR like you normally do.
9. Go into the options and change the screen resolution to the new options of 1366x768
10. You'll see that scaled menu backgrounds have been created, but they're not quite right. Ignore it for now.
11. Load up a saved game/start a new one. Holy Mother of Resolutions! The GUI is perfect! Dialogue works!
12. Close the game and start it again. On my setup, I get an appcrash if I start KotOR too soon after closing it, so wait a few seconds if you have to.
13. The menu backgrounds are properly done now. KotOR is now officially working perfectly @ 1366x768.
Here are some screenshots demonstrating it working:
FWS's developers deserve a medal. They've provided a solution to a problem that's being plaguing laptop KotOR users for years. If I wasn't a broke university student, I'd donate these guys some money right now.