We have a similar timeline to you but a couple of weeks later.
I believe you can get the supporting letter prior to applying to cic. The only info they request is:
To obtain a work permit support letter, please submit your request to PNPInfo@gov.bc.ca with the subject line: Work Permit Support Letter Request.
Within your email request, include:
your full name,
your date of birth (DD/MM/YYYY),
your Provincial Nominee File Number,
an attached scanned copy of pay stubs for the last two pay periods, and
an attached scanned copy of your current work permit.