three things:
1: battery size is not an issue. batteries are for starting, and for playing time with the key off.
that is all. Your stereo amplifier runs off the alternator.
2: that amp probably doesn't have a lot of onboard capacitance. This is one way that amp manufacturers have found to reduce size and cost in the last decade or so. They are building amps with only marginal specs on the power supply. And they rate their products very optimistically now too.
ANALOGY TIME!
When you ask them to actually produce the power they are rated for in the real world, you run into problems like this. Ie like an under powered econobox with 4 adult passengers being asked to run 75 mph up a mountain pass with luggage in the trunk might not have the power to run the A/C at the same time and keep up its speed.
Supposedly the car seats 4, can run 75 mph, can drive up a steep grade, can handle a trunk full of luggage and has a/c... but can it do all of of them at once? probably not. You need a more powerful engine under the hood if you are going to do that constantly. Or just a bigger more powerful car.
/analogy
3:
you are probably driving your amp into clipping:
you need to turn your bass boost, loudness, bass eQ etc down. those have a HUGE effect on the gain structure, and could require as much as 16x the current by artificially boosting the input levels too high before the amp.
probably turn the gain down too. not going to give you any dumb rule like "only use 1/4 gain"
What you need to do instead is
A: exercise self restraint. the knob on the head unit that controls volume? turn it to the left. I bet your problem disappears.
B: turn the bass boosting features OFF on your head unit.
And keep them off. Loudness, Media xpander off or min, etc.
C: get a DMM and set the amp's gains that way if you are still having trouble. you will probably find that you are asking 800w or more from it... I don;t know how they rate amps anymore, but when they say 500w RMS/1000w peak, what they mean now is they only put enough capacitance in them to support 500w with out causing "dimming"
So measure and make sure you are not asking more from the amp than it can supply.
a good website for you to look at regarding setting gain
http://www.bcae1.com/
http://trussinme.com/Apps/audio/voltagecalc/default.asp
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Now:
If it just isn't LOUD ENOUGH after you turn it down to perform correctly, buy a 1000 or 1200w RMS amp... and then turn it down to only put out what you were getting before.
At that point the amp will be 'loafing along' and the onboard capacitance will be more than adequate.
That is the 'easy/real' fix, honestly. Dunno how many 1200+w rms amps i have seen in a car with a 75 amp alternator that did just fine, becasue they were dialed back to put out the same wattage as the much smaller but overdriven amp (400-600w) they replaced.
or buy a vintage (late 80s-mid 90s) large footprint amp with the same power specs... they are better built (overbuilt by comparison) and generally won't have this issue.
Let me just say one thing: You only have one set of ears.
There is no replacement or fix once damaged. And damage is CUMULATIVE... even lungs will heal (slowly) after you quit smoking. Ears do not self repair.
If you are constantly (and by that I mean more than 5 minutes a week) driving around playing loud enough to have problems with dimming lights, you are playing your music too loud.
and in 20 years or so you are really gonna regret it. \\