Third time’s the charm. The first two people that I spoke to were not able to help me, and it was not even clear that they understood the problem. This time, I spoke with Tim, who clearly knew what I was talking about. This made the whole conversation much more pleasant.
While he agreed with me as to the problem and the likely solution, it was not quite that simple, since they are, of course, using some mass vhosting module. He was not able to tell me what module, exactly, they were using. Not that it matters – I was just curious. But apparently the account name makes the hostname different from what we are using, and this goes into the mass-vhosting algorithm. He agreed to change the name on the account, which should resolve the problem, but could take several hours to actually happen.
So, we seem to have a happy conclusion to this all. And the next time you call Earthlink support, ask for Tim.