Well, I called AA in Japan back, to see if another agent would let me ticket over the phone.
The agent said I could, then noticed that as I have a full 20 sectors, and therefore need paper tickets, I would have to do it at the airport ticketing desk. She also mentioned that I could use my credit card to pay for both tickets, as the surname of all travellers has to be the same as that on the credit card used! I've not heard of that before.

Is this an AA Japan rule? It doesn't really matter I suppose, as my fiancee has a card in her named linked to my account, but it does seem like a hassle, and I could see how such a rule could cause issues for some people if they didn't find out about it in advance!

I was also told that ticketing should only take 15-20 minutes, a lot less than the three hours the first agent told me I should allow, but being as they are two hand written 20 segment tickets, I think that the first agent was being a little more realistic. This of course means that I can't take the early CX flight from NRT to HKG....