Shame on you guys for not mentioning the big 3. BudK touched on one however.
A '12v' system runs at ~14-14.5v when the alternator is powering things. When a SLA is under load its output voltage will be 12.5-13.0V or so when its full (depending on load, internal resistance etc) All energy that your stereo uses comes from the alternator, full stop. When the alternators output cant keep up and battery provide current, the voltage will fall to the 12.5-13.0v or so that the batteries provide. An incan filament with its supply dropping from 14v to 12.5v will be noticeably dimmer. So no matter how many batteries and caps you add, you will still have the same problem.
The 'Big 3' improves your charging system by giving it a better path to the battery, so that it can help keep up with the stereo.
For the positive side, a heavy wire from the alternator output to the battery + terminal lowers the resistance of that, simple enough.
For the negative side, a heavy wire from battery - to chassis loweres the resistance of that, simple enough also.
Also for the negative side, you need better bonding between your alternator and the chassis. Your alternator is bolted to your engine block which is mounted on rubber mounts, and has a ground strap of unknown size and condition (how corroded is it?) to go between the block and the chassis. Get a heavy braided ground strap (you need the flexibility, even welding cable could have issues with all the vibration of an engine on mounts!) and install it from chassis to engine block (or alternator frame if there is any good looking bolts/lugs available)
A 1F cap with a 750W load (600W class D amp, allowing for efficiency, etc...) will drop from 14.5v to 10V in .066 seconds, obviously they are going to do nothing to help with dimming headlights. However, if the input voltage doesnt change as rapidly, it can allow the SMPS internal to the amp to maintain proper regulation better (I.e. it only has a current transient, to deal with, and not an input voltage drop transient also) so improving how tight/clean the bass sounds is plausible, not here to argue about that.
I would address the headlight issue in the following order:
1) Big 3
2) Check additional grounding. (where do the headlights ground? if its somewhere a lot of current flows when the amps draw a lot of power, perhaps your ground isnt good enough for them. Its called 'ground bounce' on a PCB, not sure what its called in the automotive realm)
3) Battery in trunk with hefty leads to the amp. If the voltage sags over the long run to the main batt, then the trunk battery will still provide the needed current. Without the trunk battery, the amp has to draw more current to get the same wattage, making the situation even worse. (in a severe example, if the amp is recieving 10V on a bass hit requiring 500W to be drawn, then 50A will be drawn. Vs 14.5V requiring only 35A. A trunk battery would limit the voltage sag to 12.5-13.0V during the transient, and the current drawn from the front to the rear will be smoothed some, so it would be a more constant draw, rather than a jump from low to high to low.
4) high output alternator Probably not needed for a 600W system, but it would be the next step.
5) Moar batteries, moar wire, moar alternators