Is the carbon-zinc battery that I am characterising overloaded? Using HPPC
Unfortunately not a definitive answer, however I think this explains the behaviour...

I completed an additional 9 pulses to keep testing the battery. The minimum battery voltage should be around 4.8 V as per the data sheet. When the open circuit voltage dropped beneath 8 V the load would drop the voltage beneath 1 V.

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Conclusion

Given these results I think that the test is simply not well designed for carbon-zinc batteries, which are in turn not designed for such needy loads. I do not believe that this is a result of the measurement equipment, and I do not have the relevant chemistry degrees to be able to comment on the battery composition. Thanks to commenters Uwe and Misunderstood for their suggestions and ideas.