Sorry, I couldn't understand which part of the form you are talking about. You need to fill this form to start your legal name change process in BC. https://www2.gov.bc.ca/assets/gov/health/forms/vital-statistics/vsa529.pdf and you should be living in BC for at least one year. No need for any newspaper advertisements if you are just splitting your name. I started my passport process after getting the new PR card and attached a copy of the same with the application. A few people applied before getting their PR card, though I am not sure what supporting documents they attached with their application.