I'm replacing a 200 amp residential service. Meter base feeding a 200 amp main-breaker panel that is inside directly behind the meter base. The #4 bare copper GEC from the water pipe will only reach a ground bar in top of the panel which is the service disconnct. I'll be driving a supplemental ground rod. Is it OK to run the #6 bare copper GEC from the ground rod to the grounded conductor in the meter base or must it terminate on the same ground bar as the #4 bare copper GEC.

The meter base should be fine (is),If the POCO and AHJ agree.

I would definitely check with the local AHJ to be sure, as leland stated.
Where I used to live the power co would install locks on the residential meter cans and because of this, the local inspectors considered the GEC connection inaccessible after installation as per 250.68[A]. They required it to be brought either to the panel or the disconnect enclosure in order for them to consider it accessible after installation. When the power co switched to requiring ring-less meter sockets and started using just a Buddco tag instead of locks, those inspectors then began allowing the connection of the GEC in the meter can.
Also, don’t forget, to avoid having to verify to the inspector that you have 25 ohms or less on your supplemental ground rod, you’ll probably just want to install a second rod and be done with it, as allowed in 250.56. I usually install them the length of the rod apart.

Ask the AHJ how he feels about 250.64(C) ex

You might be able to run a #4 from the busbar you hit to the one where the neutral lands.

As far as NEC is concerned you can land that #6 in the meter enclosure in this application. As others stated check poco rules. You also will need two rods per NEC.