When I'm prompted for Disc 1 again after installing the data from Disc 2, I insert Disc 1, and once it spins up, Windows (XP Pro SP3) gives me the Blue Screen of Death™ and restarts, apparently with no registry values written so I can't run the game. Google has basically failed me; all I get there is issues with crashes after install.

Anyone know what my problem is, and how to fix it?

write down the error codes of the blue screen, 0x...... or something.
then you can google on what that problem could be.

maybe your cd is too scratched?

Smells to me like some problem with Disc 1 (unreadable section or something like that)

It's probably less work to simply pirate the game and use one of the images as a disc replacement.

This wound up working, but unfortunately it turns out I can't run it because my video card doesn't support vertex shading. FFFFFFFFFFFFFFFF

If you are running an Intel GMA 945 or higher you can use a vertex shader emulator, just have a read of the FAQ etc on it :

http://angelictears9xxssf.wordpress.com/

Curses! Vista and 7 only. Now I'm really starting to regret opting for XP Pro when I got this laptop.

You could try 3D Analyze, that emulates pixel and vertex shaders and supports XP with Service Pack 2 or higher :

Instructions :

http://www.raymond.cc/forum/tutorials/11575-3d-analyze.html

Info :

http://crystalunicorn.wordpress.com/list-of-games-that-work-with-3d-analyze/

Download :

http://supersid2006.tripod.com/id4.html

You have to click Download tab then scroll down a bit. Its not 100% guarenteed to work but worth a shot as Republic Commando is on the list.

How would I make that program work with Steam apps? If I can make it make Republic Commando work, I'll try it with Battlefront 2, which I own on Steam.

I have never had to use it but at a guess you just have it running with the relevant profile in the background. From what I understand from reading there is a game list and you select your game from the list, then just follow the instructions from the link I posted above to select and setup the relevant shaders. Then just minimise it so it is running in the background and start the game.