OK, so I got a new battery and now I can get to the fastboot screen. However, I can't reboot into anything; all I get is the white "HTC" screen for a few seconds, then the screen goes dark, and ten or fifteen seconds later I get the "HTC" screen again, etc. etc. etc.
The HBOOT version was 0.98 something. I was able to run the CRC check, if that information is useful.
I can't leave the battery in the unit because it will continuously reboot and the battery eventually starts to overheat (I'm guessing that's what happened to the original battery).