Ok, I managed to fix my phone.
The first thing that differed about my phone from the video tutorials I watched was that when you are preparing to flash the phone, you're supposed to hold Vol+ and Vol- then connect the USB cable. In the video tutorials the LG logo came on at that point, but on mine it did not. I figured out that if I plugged in the cable first, then held Vol+ and Vol- the LG logo would come on. This was wrong. The P999 does not display the LG logo when ready to flash. Hold Vol+ and Vol- and then connect the USB cable, wait a few seconds and then let go. Your phone should be ready to flash now. If it's not, try again.
Secondly, this is the software you want to use to flash if you've bricked it: T-Mobile G2X One-Click Recovery Flasher. The version I found on some other websites was outdated and wouldn't fix the problem I had mounting the SD Card (which is actually internal memory on the G2X). With this version I was able to boot into Recovery mode, mount the SD Card manually, and then flash the ROM. I was able able to properly enable USB Mass Storage from recovery mode, which would allow you to copy a new ROM to the SD Card if you've somehow lost your data.
The final issue I ran into was no Google Apps, and no ability to sync my phone to Google. You can grab the Google apps and Flash them through Recovery mode here.