I finally got the emulator to work on my DSi. Basically, it seems like there are (at least) two different versions. Initially I downloaded the file from http://www.dsiplayer.com/
This version did not work, and nothing I did would get the emulator running. However, I then tried downloading a different (older?) version from here http://www.zophar.net/consoles/nds/gba.html
And everything worked fine, just dropped the plugins folder into the system folder, put it in my DS Lite, and everything was running fine. I have no idea what the differences between the files are, but it seemed to be enough to make one version useless on my lite. Maybe it's a firmware issue, or maybe it got bugged somewhere along the line, but the one from Zophar hasn't given me any problems.