I've got it down to a OWE4, with the SYD-JNB route eliminating Asia. Here's what I'm looking at now.
ATL-DFW-HNL
HNL-SYD
SYD-AKL
AKL-CHC-ZQN
ZQN-CHC
CHC-SYD-JNB
JNB-CPT
CPT-LHR-ZRH
(overland to Berlin)
TXL-LHR-YYZ-ORD-ATL
I think this should be valid. I have only four stopovers in Oceania (Sydney, Christchurch, Auckland, Queenstown) though the CHC-SYD-JNB flight has me overnighting in Sydney just from the schedule's. The stop's only about 16 hours so I think that just counts as a connection. (Same thing with Toronto, actually, I have an overnight stay there on the way back but an extra stopover left in NA anyway.)