1st, there is no lhr-bne direct flight. I would skip the HNL routing given the limited amount of time that you would be there.

Consider this revised routing : bos-mia-lhr-dxb-lhr-syd-bne-chc-syd-jfk (36,356 miles) followed by jfk-lax/sfo/yvr (transcon)-dfw/ord-bos (approx another 5,000 miles)

The syd-jfk flight (QF 107) is only 1 flight and does not count as the 1 transcon in North America. This leaves 3 segments for North America and 1 stopover which can be taken after you return to bos (via a separate bus/train/plane) from QF 107. QF 107 leaves on Sunday 2-Nov and arrives the same day in JFK. A less hectic schedule, IMO since the routing to australia is going to be very long and tiring.