We have received a letter saying the evidence we have provided is insufficient and a chance to show more. Is this the same as the Procedural Fairness Letter?Your danger is in the in-between. It takes some time to get the application refused and CBSA is notified immediately; you then have to gather everything (has the police certificate expired, etc) and reapply, and then it takes 6 months or so for IRCC to acknowledge that you've actually applied.
Please focus on proving you live together. Fretting about how to re-apply and the public policy against removal before you've even received a Procedural Fairness Letter warning you that your application is likely to be refused without new evidence is not a good use of time.
I understand what you have explained regarding the time frame. If the CBSA is notified by a lawyer etc that you are reapplying, is there no way to start the process and send a new application?