As we are now living in Mexico, I thought I would answer to outline what ended up happening with this.
We applied for Resident Status (Visa de Residente Temporal) at the Mexican Consulate in New Zealand. When we went in for our interviews I discussed the work situation with the Mexican Consul. She made it quite clear that we did not have work permits, and that we could not work for any Mexican employer and get paid in Mexico. We could however continue to work remotely for a NZ (or other country excluding Mexico) employer, as long as we were not paid in Mexico. If any payments were made into a NZ bank account then it was quite legal and acceptable.
In fact she told us that it would help us in the long-term to have this income when we reapply for our Temporary Resident Visa (after the first year) to satisfy the economic solvency requirement. As we are primarily in Mexico for voluntary/charity work, she said donations and sponsorship might not satisfy the economic solvency requirement (irrespective of the amount) as donations can stop any time without warning.