Sadly QF is the easiest one from the online tool, but comes with it's downfalls for certain. JL would probably book it but has a service charge in addition to any change fees you would normally incur.

I'd say in your case a travel agent would be your best bet and ticket on QF stock (assuming you have any substantial AA segments) as in my experiences QF ticketed RTWs have AA segments credited as distance while my CX ticket did not (credited as revenue at about 1/3 the rate I should have got on a distance credit).

BA apparently does still do ticketing by phone call, but not sure their requirements for BA codes on the ticket. Unfortunately on an ex-Japan, going with CX is likely not an option if you're using JL for your overwater (can't return to the country of origin until you wrap up the ticket). The way I got around that was taking CX to Oceania and a QF code from there to the US.

I've heard also that UL provides a decent experience on these tickets, but that's from someone else in this forum. I haven't flown them for about 10 years, so no firsthand experience at all. Their own coded flights credit the worst of any OW D fare in the network (to AAdvantage), so I have little motivation to fly them.