I just did a relatively last-minute issue ex-NRT. I called the AA RTW desk, had them set up the booking, and then called the AA Tokyo office for payment. I called AA in the European afternoon (morning in Dallas, just after they had opened), and then Tokyo a night later (morning in Tokyo). The ticket was issued less than 36 hours since my first contact with the AA RTW desk, but I think you could do this even faster. The trick is to make sure that AA has time to price the ticket (I seem to recall they live in Dublin for pricing issues). Once that is done, Tokyo can issue it. Utterly painless process as far as I was concerned.

I would echo the comments by JohnAx above to avoid a complex ticket and to keep it uncontroversial. Better get the ticket issued fast and start flying. You can always reissue later with additional and more interesting segments.