Since it is not mandatory, I think we can start to work with our SIN, while amending it. I asked some employers and they don't ask for a study permit just sin number is ok.I have exactly the same issue as you, I called IRCC and they told me that it's better if I change it. but I consulted with other migration lawyers and they said that it should be fine, since we still meet the criteria to work off campus (Full time master students at DLI institution). I don't really know what to do because it looks like amend the study permit could take up to 6 months.