Since its a service entrance panel, the neutral/ground is bonded, so I dont see any issue with changing to a different ground bar.
Do you understand what the guy means when he says the neutral/ground bar is bonded? What that means, and yes that is the case with the meter/Load Center combo unit you bought. The service neutral conductor is bonded, connected, to the meter socket steel enclosure. Inside the Load Center electrical panel the neutral/ground bar is bonded, connected, to the steel metal enclosure again. Yes that is the way it is done. Even where the meter socket is separate from the electrical panel it is Bonded in the same way. (At the meter socket and again at the electrical panel.)
So can / does the unbalanced load current travel on the steel enclosure between the two bonded points as well as the AL bus between the meter socket main service bonded neutral connection and the bonded neutral bar connection? You bet.
What you should have asked the guy, is the AL bus that connects the main service neutral conductor in the meter socket to the neutral/ground bars in the electrical panel made of 6061-T6 AL alloy? That is what you need to bolt a copper neutral/Grd bar to the AL bus. If the AL bus is not 6061-T6 AL alloy and you connect the AL bus to copper the connection in time will fail. Will the bonding screws/straps/steel enclosure carry the unbalanced load current back to the utility transformer? ya, I guess.... Will the bonding screws connections to the main service neutral conductor keep the L1 to neutral and L2 to neutral 120V nominal voltage stable? Hopefully.... Will the total resistance of the bonding screws and steel enclosure be less than the AL bus that connects the main service neutral conductor to the 6061-T6 AL alloy neutral/Grd bars? I doubt it...
.