I am in Japan 3-4 times each year now on long stays (1 -2 month per stay typically as I am learning Japanese).

I was always able to find Avios flights with JAL or AY in J. I normally book these last minute few days to a week or so prior to departure and never had an issue. Last JAL flight was to FRA then separate avios to LHR. For AY, I was able to find redemptions via HEL.

For cash bookings, I flew below £2k via Doha (QR, Q-suite) then BA to LHR (Club Suite, same ticket) for about £1900, just a couple months ago.
That was expedia fare, didn't show on BA.com and included a J segment from FUK to ICN on Asiana, all one ticket.

I seldom fly to Tokyo nowadays as I prefer to explore more interesting parts of Japan like Yakushima, or even Kyushu by rental car. I highly recommend rent a car. Last flights were to FUK via ICN (see QR above).

In Japan, Shinkansen is convenient for travel between cities but to see more rural and (in my opinion) authentic Japan is it just more convenient and time saving to drive. Certain nature places are very inconvenient or time consuming by bus/train. But if you there for big cities, Japan has one of the best train systems in the world.
I seldom book months in advance now because I am there frequently but never had an issue finding good fares at relatively short notice.