As a follow up..... I was replying in the spirit of your remark that "I'm pretty flexible on the routing".

Starting from Japan would seem pretty feasible since I imagine the real starting point is LAX/SFO. If you started from there and first flew to Japan to start the RTW there, continuing west or south you could end the RTW in SFO/LAX and not fly the last leg to NRT. That way you would only need an additional one-way to Japan.

Although if you don't want to waste that last leg you could buy a return and make another trip out of that. Remember, you have a year to fly on your RTW ticket. So you could also do the positioning flight(s) long before or after the rest of the trip. And you can be very creative with the route...

Another great use would be (assuming starting in Japan) to get back to SFO/LAX and save that last flight back to Japan to make a stop in Hawaii. Have another vacation there and fly home (either using what's left of your RTW via NRT or with a one-way from HNL). Your options are plentiful since you can have 16 stops with 5 of those in the US (a connection is also considered as a stop, so in practice you will probably use a few stops on those).

Using miles to do the positioning trips is a great option. They are flexible and with a lot of programs you can do one-ways. Your RTW will earn you a decent amount of miles so those might be almost free as well.

And the savings really are substantial, if you go for the most expensive option (up to 39.000 miles) you will save about $9.000 per person. That should be plenty to buy a few additional flights.