Oh wow... Your expectations are much too high. First, that link is not a battery, it's a capacitor.
To realistically run a 1,000 watt PA for 8 hours, you are going to need some BIG batteries. I have a 100 lb, 134ah C&D Battery in my truck. Tests have shown that those batteries can handle a 2000 wRMS tone for 20 minutes before reaching zero capacity. Off music, you're looking at an hour at most if you are cranking it.
This is something that can be done, but I wouldn't go over 200-400 wRMS if you expect any kind of extended playback. You'll still need to fill your cart with like 300 lbs of batteries, though
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