Then yes, you should tell INZ.
I don't think it will affect your application in any strong way, because almost everything in an application under SMC hangs on the main applicant's qualifications and experience, with a small number of extra points allowed if the partner has a qualification, recognizing that another educated person available for the NZ job market is a good thing. So the overall impression made by this offer of a place will be a good one, but without there being any advantage (you've already got your ITA, so any points due for your partner's existing qualifications will have gone into that).
If your partner's course will begin before your application is likely to be processed and finalized, there will have to be a separate application for a student visa in his/her own right. That won't interfere with your application under SMC.