Went through inspection this last month with a dead TPMS sensor, passed no issue at a dealership. It's not a requirement and you can google and find the handbook for techs.

Depending where you go, someone may play fast and loose with it and try to get you to pay them to replace them saying you'll fail otherwise.

Spoofing a phone number is incredibly easy and caller ID shouldn't be a form of verification. Call the main 800 number for your insurance company. Ask them if it's legit.

Insurance did an outside inspection when I bought my house, but never after. If you didn't change companies I would call your insurance company to confirm it's legit, then ask them to confirm the inspectors name. Could easily be a scam or something.