Your 7th BA flight (if codeshares count) comes from taking the NRT-SIN codeshare rather than the JAL code (I am assuming the null entry in your itinerary for a BA sector here is not a typo'). Inventory available in D only, not A, bizarrely.
Your 8th BA flight will have to come from a North American codeshare. Why not alter the end of your journey to:
PIT-JFK-SFO
SFO-LAX/SAN (i.e. a surface sector on to San Diego)
SAN-ORD-LHR-TLV (three sectors, including the BA codeshare SAN-ORD)
Does that work?