A "Grounding Electrode Conductor" (i.e., the wire attached to the ground rod or other electrode) never has to be bigger than 3/0 (copper). That is in Table 250.66.
An "Equipment Grounding Conductor" (i.e., the wire that connects the external metal parts of a piece of equipment back to the Ground Bus at the main service panel) is sized, using Table 250.122, on the basis of the overcurrent protection device for the branch circuit or feeder that provides power to that piece of equipment. In order to require a 350 MCM EGC, the overcurrent device would have to be at least 2500 amps.
So I am not certain what is going on here. To what is this mysterious wire connected at each of its ends? If it's an EGC, where is the 2500 amp overcurrent device, and to what is it supplying power? If it connects to an electrode, then why does this engineer think that a 350 MCM is needed?