The problem is voltage and amperage. The cap is designed to keep voltage there during quick voltage drops in the electrical system. However, it recharges itself from the charging system (Alt. and battery) and those slow bass notes need current from the alternator and a good high amperage / deep cycle battery to stabilize it. The problem is that a high output Alt. is pretty expensive and most people dont spend the $. You need to have the charging system checked to see is the Alt. has been damaged from the abuse. Make sure you are using high quality cable and that the connections are clean. Battery terminals and grounds should be clean from debris and the ground cable of the amp should not be longer than 18".
If you can afford it, Get a blue top Optima and have the alternator upgraded at a local alt. rebuild shop. They can tweak and upgrade them pretty reasonably. Good luck!

P.S. The battery voltage in the car is probably dropping below 10volts and that is where most head units shut off like your is doing. It also affects the ECU voltage which affects performance. That is why compnaies like Apexi etc. sell all those grounding kits. They ensure the engine is getting good grounds for maximum output.You can test this by using a digital volt meterand watching the voltage at the radio when the bass hits.