An alternator recharges a battery. The older alternators would work at full charge and dissipate the excess to keep it in range. If your alternator wasn't producing enough, you would see your voltage steadily dropping. It might just be going to slow to notice, but I would expect it to take at least a half hour to an hour on full to notice if the alternator is not keeping up.
I know your vehicle has the computer controlled alternator, and I think you do need to upgrade your alt, especially if you fix the upcoming problem.
A battery is going to give you a larger storage bank to pull from. This is on the same principle but you are starting with more stored energy. If your alternator is not keeping up, you will see it drop, it will just take longer to empty. The larger storage bank is really only best if you plan to use a lot of that energy when it WONT be recharged, (because the engine is off). If you use it for work, or outdoors, or need the energy from the vehicle for an extended period of time, then a larger battery is better.
I think you will need to upgrade the alternator when you fix the first problem which is: 0ga wire is nowhere near big enough for a 4K watt amp. 4k watt amp /12v = 333 amps. Add that to your nearly 100 amp vehicle with everything else, and Your Big 3 and your amps main wiring should probably be at least 3/0, and I would probably go 4/0 just because. If you are cranking up a 4k watt amp and have 0ga wire, you are pulling amperage out of your vehicle's electrical system.
It should be the first thing, but I would start with redoing the big 3 you just had done, with larger wire, (My suggestion is 4/0, I've had good results with Knuconceptz, or whatever you choose). Your amp's power and ground wire should be at least 3/0.
After that you should be pulling much more amperage, and If that's the largest alternator you can get, get that one. Ideally an alternator that can produce the amperage of your entire electrical (lets say 333 + 100) would be enough to keep your batteries from draining. It might be worth looking up to see if dual alternators at that size is an option. If someone made em for the previous model it might fit, or you might help show them there is a demand for em. If you have multiple options at that size, you want to know two things about that alternator: what is it's amperage output at (or near) idle, and what rpm is needed to produce max output. Any quality manufacturer can give you at least those numbers, and the better ones can give you a curve to that exact alt when they test it. Compare max output to what RPMs you are usually driving at, and see if you're going to be able to charge.
If after that there is currently not yet a larger option, THEN a battery with a larger starting bank would be the third thing to upgrade. This would probably be somewhere in the range of listening to it at full volume for more than an hour.