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Well, I *finally* got in touch with someone at the DMV who knew what the heck she was talking about. It turns out the original agent's 2-4 weeks estimate was a work of optimistic fiction; this agent said it would take at least 90 days and probably longer.
She did say it sounds like the letter was correctly worded so once they process it it should all work out, but right now the letter is most likely in a massive Inbox that was being worked through on a "first come, first served" basis so I shouldn't expect to hear anything until May or June.
They're not technically allowed to give me an operating permit until then, but she wrote up a case on my behalf and is asking the field office to make an exception for me.
So at least progress has been made.