When I do the test challenge for either the Dragon or Lo Pan, I get a few fights in and the game crashes. Every single time.

I'm using the Restoration Project v2.2.

Anybody else have this problem and know of a fix or workaround?

Let me guess, you have saved after a fight?
If i remember correct your only way out of this is loading an older save and than don't save during this fight.
SF is the most bug ridden place and Fallout 2 is best played with several different saves.

Actually, no. I've saved a few times before I even talked to Lo Pan or The Dragon, but was terrified of messing with anything during the fight (seemed fragile), and was hoping I would get through to the end.