When you say "adding a cap is only going to initial pull from the battery", I agree in part if the car is a) not running and b) that the alternator could not supply enough amperage to the system in the first place. Then and only then the the reserve of the battery's power is use to compensate.
The moment that the car is running, the battery becomes a load to the alternator. The battery cease to becoming the source. If the system demands more than the alternator can supply, it will draw from the battery. If continue long enough, the battery will be dead.
By your rationale, the battery is the source of power and if that is the case, an alternator is not needed and your car can run. BTW, a car can run indefinitely without the battery after being started but cannot do so without the alternator. FYI - practically every amplifiers uses capacitor in their output stage, digital ones does it slightly different, but that can be left for another day. All that said is that if you have an amplifier, you already have a fast switching capacitor in your system.
Although the battery, alternator and in part, the capacitor all does essentially the same thing; supply power once charged or rotating. They do it in however in very different ways.
Alternators are designed to supply constant power at certain RPM. Contrary to popular beliefs, the car electronic system is not 12 volts but it is 14.4. That is the charging voltage of the alternator.
Battery are designed to provide a buffer to the system and when the alternator is not in use. aka (starting of the car)
Capacitor in this context is to supply transient power.
Music by definition is not a sine wave where it is constant. If that is the case, a capacitor would be useless. But because music has peaks and valleys, while the load is low, the alternator has already charged the capacitor AND the battery and on on the peaks (dimming of the lights), the capacitor would supply the fast spike past what the alternator is capable of supplying aka dynamic headroom.
There is way too much to get into the specifics but the bottom line in relation to the O.P question, a capacitor would help but has dependencies of other aspect as well. I agree that an high output alternator is the way to go, the addition of a capacitor has it's merit especially in raising the peak capacity of a system which essentially from what I can tell from O.P. is the issue.
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