First off, you are guessing at 1800 watts. Second, even if your 1800 watt (average) guess was correct, at 12 volts that's 150 amp current draw, on a stock alt that is likely less than that, and also must support the car itself. You are trying to convince us that a stock charging system/alt can support a 4kw system due to transients and imp rise, this just tells me you have little to no experience with such a situation. Third, adding more batts to a system whose power supply is insufficient will only prolong the inevitable, dead batts.
Impedance rise does not affect input current draw for the amplifier, only supplied power to the speakers themselves.
Extra batts will help the situation, but it will still require additional charging with the stereo not playing. That means either regularly driving without the stereo on, or as others suggested, charging the system over night with an external batt charger.