FYI, Arturia support says it's a bug they haven't seen before, and got back to me this morning with a list of things to try. I should say---the problem is frustrating, but for all the complaints about slow responses I've seen on these boards, their tech support has been in touch as promised.
(In response to your questions: Yep, tried disabling the firewall. And like I said, I can get into the website, see my registrations, etc. So the account is OK.)