Without anything to cite, I would say the procession is specifically the process of transporting the body from the church to the cemetery in a parade-like manner with police escort.
Oh wait, here, I found it:
Well there ya go. I guess you can have it in the church and at the cemetery, but you cannot have it with you while you're traveling in the procession as specified above. The easiest way to stay legal in this case would be to unload and lock the firearm in your car during transit, therebyremoving the "deadly weapon" status.§ 20‑157.1. Funeral processions.
(a) As used in this section, a "funeral procession" means two or more vehicles accompanying the remains of a deceased person, or traveling to the church, chapel, or other location at which the funeral services are to be held, in which the lead vehicle is either a State or local law enforcement vehicle, other vehicle designated by a law enforcement officer or the funeral director, or the lead vehicle displays a flashing amber or purple light, sign, pennant, flag, or other insignia furnished by a funeral home indicating a funeral procession.
I hope that helps. I'm sorry for your loss.