with the requirement that I have to be in the job for 3 months in order for INZ to process my residence application
No, this is not correct.
https://www.immigration.govt.nz/opsmanual/30614.htm
As soon as you get the offer of a skilled job, you go to the nearest INZ office with your evidence (letter, contract with all the small print), and this will be sufficient to have you granted the Residence that is waiting for you. Your Residence visa will have a Section 49 (1) condition on it, that you must work in the same job mentioned for three months. When you can show evidence (pay slips, tax and financial records) that you have done this, and are still working at the same place, you go back to INZ, and they will give you a new Residence visa label with no conditions. After that, once you have your Residence free and clear, you are free to change jobs if you wish (subject to giving notice in line with your contract, of course).
To be clear, then, you need NOT wait to have been working for three months in order to get your deferred Residence granted.