Seems the problem BA London has is that I did not book specific flights and dates so they could not determine that I was in transit AKL-HKG-LHR and that I was only making one stopover in LHR, the other two passes through were in transit. Thus, I only had a total of two stopovers in the continent of origin.

I told BA CAI that I would book actual flights if I had to but that I had indicated the transits with the "x/o" indicator on each segment. No word back from BA London yet so I am on hold until Monday.