I thought the truck was falling apart (like mine). Then I read 1000w and about fell out of the chair.
OP, GM trucks utilize the vast majority of their alternator current just running the truck.
In order to solve your dimming issue, you at least need a second battery as close to the amp as you can get it and the big 3 upgrade. You might still run into problems, which is where a high output alternator, or a second stock alternator specifically for the stereo would come into play.
Also, on the regulator thing: Somewhere in that neighborhood, GM started doing a fuel saving technique that actually kills the alternator and runs the truck strictly on the battery. Voltage will go from 14.5>12.5 when ever you aren't putting a load on the motor. If yours has this, the only way to make your alt charge the entire time is to get an external regulator to override GM's fuel saver attempt. I know older trucks didn't have this, I know newer trucks do. I am unaware which of those two categories yours fits into.