The thing with capacitors is that they're useless if your charging system isn't up to par. In this case, putting a cap there wouldn't make a difference since, to put it bluntly, the capacitor isn't being charged fast enough as it is being discharged.

Besides, you don't need a monster cap. Rule of thumb is 1000W for 1 farad. Going more is like putting gauge 0 speaker wires in your car -- a complete waste.

It's a throwup IMO.. a less expensive way is to buy an extra battery. This should allow you longer playing time especially when you're simply parked with the engine off. I also think it would remedy your dimming lights. However, your alternator is faced with another load to charge.. hence I can't simply recommend throwing an extra bat. A high output alternator seems like your best bet here.