Deep-Cycle Battery or....?

Hi, I'm Jezer and I got two Alpine MRP-M500's powering two Alpine 1243d's wired at 2 ohms load producing 500 rms each. I have a '98 Dodge Stratus and my stock battery and alternator is 90 amps so obviously it ain't powerful enough. I can't replace the battery to a yellow top or any deep-cycle battery because none of those batteries fit in the little space provided for the battery in my left front tire. SO my question really is, could i just get a higher output alternator and keep a stock style battery? If so, how much would amps would I need? or that won't work?