Sticking to "
F [that] Avios has access to", I think the best is JL. JL F also has close-in availability surprisingly often on LHR-Tokyo though 3 seats could be a struggle; 2 is more achievable. JL Business is also quite fine.
CX is quite good, followed closely QF.
JL F service is very good. Extremely thoroughly attentive crews who practice the 'invisible elves" style of service - you're expected to use the call button at which point one will appear next to you within seconds. The ground service also does not suck, for example the gate agents will tend to find you in the queue and take you to the front. Clearly, They Do Know Who I Am
