You definitely cannot be in NZ while applying for Silver Fern.
For the next part of your question, this is just my guesswork, and
you would need to ask INZ for a ruling. I GUESS that it wouldn't work to just pop out of the country for the day of the application, because although that part would presumably go through if you were quick on the keyboard and lucky, as soon as INZ got back to you to get your evidence, your contact details, IP address etc., would show that you were supplying this from within NZ. I'm assuming that the restriction, whatever the reason for it (which has never been explained as far as I'm aware) is on the whole process, not just on the application.