When a car audio amplifier goes into protection mode, does that mean I need to add another battery, or capacitor, or what?

1000w of amp power actually isn't that big of a load on your charging system.  

I've routinely run ~1500 on stock alts and old batteries and never had short term problems.

My guess is you have some resistance somewhere in your power/ground path causing enough voltage drop to trigger protection.What gauge is your power/ground wire?  Where is your ground connected?Are your wire terminals well connected?  Wires in/out of your inline fuse.Any bad connection can cause significant voltage drop.

In addition to the possible low voltage issue, which "DVC Kicker 10's"? (i.e. what is the model number or individual voice coil impedance?) and how did you wire them? We cannot take your word that they are wired correctly. If they have 2 ohm voice coils and you've wired them coils in parallel, and speakers in parallel then you've created 1/2 ohm total impedance which the amplifier cannot handle. 

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Before you waste your money on a capacitor, read this: https://axleaddict.com/cars/Car-Audio-CAPACITORS-W...