Change in passport surname being dependant

[Berta Garcia]

Hi!
I'm currently living in Canada, with an OWP because my common-law partner has a CWP as a doctor here. I want to change the order of my 2 surnames (in Spain we have 2). That would mean a change in my legal name, therefore passport and work permit. As my work permit is dependent on my partner and our common-law certificate has my current name, can you guide me in how should I proceed to be sure I won't lose my OWP once I change my surname?

Thank you!