Then why would the ECM open air valve and send gas at idle to run ac? Point being, it only sends so much air and fuel to change engine rpm's a bump over the 800, because the idle rpm of a hemi (maybe a ******) is not enough to power the AC, it's a fact that is why the ECM sends more fuel and air. Then there is actual current needed versus what the ECM sent in boosted idle. It takes that extra from the battery. It still powers what is send through the battery, and also compensates what isn't coming from the pulley. The battery compensates in general if needed, it is sacrificial to the act of turning off the engine, the alt and battery and engine work together to keep the engine on. The battery remains in the circuit with alt, the demand of the entire system at idle is too much and now add a bad battery to this balance.
Once the engine is running, the alt takes over, providing electricity and charging the battery. The battery also acts as a voltage stabilizer for the electrical system, acts as a reservoir for power demands. In this role it can deplete itself especially if capacity is low from age on battery.
Hopefully OP can clear up the question on battery.