In California there is a DMV statute that exempts Model A commercial vehicles. including pick-ups, from being required to have commercial plates.
However, if you go to a California DMV office more than likely they will tell you that you need to have commercial plates and will send you off to have the vehicle weighed.
For that reason, you need to take the attached document with you when you go to the DVM to be able to educate them. It is also best to carry the document in the pick-up in case a cop stops you and says you need to have commercial plates.