Get better grounds, do the the big three upgrade. Get better terminals. Thicker wiring. Get a capacitor. Get a second battery specifically for sound. Those are basically your options
If you want to solve your issue, get a sufficient battery capacity to dedicate to the audio system, and use the correct type of battery, then finish it off with a bigger alternator .. 14 volt batteries wont do it for you.
The alternator charges at 14 volts in order to sufficiently charge the battery .. With the car off, you should have a reading of 12 volts regardless. I'd assume that if you were to charge the battery at 12 volts, the car would eventually die while driving it because it wouldn't be getting a sufficient enough charge to power everything .. who knows ..
The car can handle 14v like others have noted but much over 15v will cause havoc with the OEM electrical systems. I had a customer a few years ago that was having some issues after we just installed a Sony radio. After some digging we found the alternator was putting out 15.5v to 16v! The excess voltage was causing the radio to shut down and eventually fried the brain of the car. Thankfully the dealer was local and took care of the customer and replaced it under warranty.
Will this happen to your Accord simply by swapping to a different battery? Probably not but I would be careful when it comes to modifying the charging system. I have two batteries, three amps, and all of the extras that go along with it and still have the original OEM alternator powering it, with 230k miles on it. The best advice is to never listen to the radio with the motor not running. Lowering the static voltage of the battery will lead to diminished capacity of the battery and the alternator will be forced to overwork to try to charge the battery causing the alternator to fail prematurely.
are you on stock alt and stock front batt? first, check your grounds. make sure all paint is sanded away to make a strong connection. i would ditch the cap, it puts unnecessary strain on the alternator and battery. check the grounds on your big 3, again, make sure the grounds are solid.
as far as the battery in the back, make sure your front battery is in good health. remember, your system is as strong as the weakest link (being the front battery if bad) you can get it checked out at any auto parts store.
how many watts RMS are you pushing? have you clamped your amp?