Well, I got a couple of emails from customer support this morning:
Thank you for contacting us for assistance with properly moving characters between worlds on your account. We understand that there are limitations and congestion issues, and this can cause a number of troubles we would like to help you resolve.

Having reviewed your request and your account, we will be able to assist you with transferring your characters according to your request. However, we would like you to understand and agree to the following restrictions before we process this request: [the restrictions they originally listed with world transfers] If you agree to these restrictions, and have made any corrections to meet them, please respond to this ticket or send a new ticket noting your agreement, and we will begin the world transfer process. This process may take several days to complete from the time you submit your response.

Thank you for your patience and understanding, and we look forward to hearing back from you.
Final Fantasy XIV Support Team
and I guess they considered the second support ticket I'd made a couple days earlier an agreement response:
Thank you for agreeing to the restrictions related to world transfer.

We have submitted the characters for transfer, although this can take several days to complete. Please be patient with us while the request is processed and we attempt to transfer the characters as you requested.

Final Fantasy XIV Support Team
So if this works out, than thanks for finally coming through XIV Support. I'll update my post when I see the results.