It seems to me that being the other 120 is already treated it wouldn't matter
correct. there is no need to add conditioner for the old water in your tank. just the new chlorinated water going in.
now, Prime is a bit different, because it is supposed to neutralize ammonia and nitrites, so if you had a spike in your tank, THEN you have a good reason to dose for the whole tank.
but once your tank is cycled properly there should never be either of these spikes..
I don't use Prime (yes I am in the minority) because every thread I see about it says it makes your fishroom smell like a giant fart after dosing.
and I don't even want to go there.
I use Aqua Plus (a Hagen product). been using it for about the last 10 years.
before that I used to use the Wardley's plain dechlor..
Prime will save you a lot of cash in the long run. Switched three years ago and I never looked back.