I did some rerouting and ended up with the following

CAI-LHR (BA)
LHR-ARN (BA)
ARN-LHR (BA)

LHR-HKG (CX)
HKG-SIN (CX)
SIN-HKG (CX)
HKG-TPE (CX)
TPE-NRT (CX)

NRT-JFK (AA)
JFK-LAX (AA)
LAX-DFW (AA)
DFW-ANC (AA)
ANC-DFW (AA)
DFW-BZE (AA)
BZE-MIA (AA)

MIA-LHR (AA)
LHR-CAI (BA)

That is ok, right?
Then I don't have to fly IB either (Guess F with AA is at least slightly better than with IB)
Further, Does arriving from an international flight (NRT-JFK) qualify for flying QF JFK-LAX?

Best