Update: I was able to order both bands just now even though one of them wasn't in stock an hour and a half ago.
Meanwhile, here's some info for anyone else ordering their bands. On my order, the shipping address listed was my sister's address, not mine. I assumed this was the case because she's the lead on the reservation, so I also assumed--wrongly--that that meant that her name would be on the shipment. Wrong. But I didn't find this out until after I completed the order, when they finally listed the entire address--my name + my sister's address. We don't live together, don't have the same last name, and don't even live in the same city.
I replied to the email that contained the order confirmation, however, I suspect it's possible that no human being will ever see it, since the confirmation email was probably sent by a bot. So then I did the thing that Disney makes more difficult than anything else: I called them. Really, I'd rather wait in a 2-hour line at FoP than call WDW. But I did and after nearly 30 minutes spoke to someone who said he'd take care of it.
But at least we got the bands we wanted: 1 It's a Small World and 1 Minnie dots.
Good luck to everyone else involved in this process!