I just talked to Service Canada about switching from temporary to permanent SIN and wanted to inform you guys in case someone finds it to be helpful.
Basically, temporary SIN is still valid until expiry date and can still be used for filing taxes, etc. as per usual even if you have PR now. I asked if I can apply for new SIN after I file my taxes for 2021 to avoid any confusion with the CRA and they told me yes, this is doable.
So, the bottom line is, as long as the existing temporary SIN is still valid, you can apply for the new permanent SIN any time you like, and then inform your employer, bank, etc. The important part is that once CRA obtains the new SIN information from Service Canada automatically, they cancel the old one and My Account with CRA will not be accessible anymore. So, you will need to register again with CRA with the new SIN. The whole process from applying for a new SIN with Service Canada (online or in person) until CRA updates the SIN takes about a week or two.