From the email you've posted, it seems that your photo ID and the approval email is sufficient, assuming it came from someone at @burningman.com. I do find it's odd that they didn't provide a confirmation number or some other info, but assuming that you're in the system it seems you'd be on record. I am surprised that they're taking cash/money order at the gate, only because it might make the other people think that they could buy tickets there too, which they can't.
If you haven't already -- and I expect you have -- email all the addresses related to conclave ticket sales (email@example.com
Also see if your approval email is being displayed in HTML mode with loading images instead of plain text mode. Maybe some graphics have the number or other identifying details in there? Worth a shot.
Beyond that ... well, I guess you'll have to take the trek out there and see what happens. Print off the email with the full email headers so that you have as much proof as possible. Bring your ID, bring some cash.
Be prepared if it doesn't work out.