My correspondent in NZ received Martha's birth certificate from GRO a few days ago. Attached is a heavily cropped version. We were correct that Martha really was born on Slains Castle on the way to NZ. The details given here were apparently very helpful when filling out the request from GRO.
Amazingly, it seems the ship left Deal on 18 Sep 1840, and Martha was born just four days later, on 22 Sep 1840. One can only imagine how her heavily pregnant mother felt as she stepped aboard.
If anyone wants a full copy I can ask my correspondent.
Topic can be re-closed now. Thanks for all your help.