Stan's right - a regular car battery won't go a month with abuse like that and it'll crap out on you. You might be able to get it replaced once under warranty, maybe twice - Third time they'll see the pattern and start asking lots of questions.
You MIGHT be able to get 3 or 4 months out of a battery if you use a Deep Cycle rated Trolling Motor/RV battery, but they aren't very good at providing the big bursts of power (200 - 400+ Amps) needed to crank a starter motor over.
Trust me, the 1 - 2 HP that the alternator takes won't make that much difference in the long term.
Believe It Or Don't: Burning the gasoline to spin the alternator directly is actually far more efficient than using Utility Power to charge the battery at home.
Between the thermal losses incurred with a steam or fossil fueled generation system, step-up and step-down transformers and transmission line losses, you lose almost half the energy you started with through making the electricity and getting it from the power plant to your house.
Hydroelectric, Wind, Solar is a lot more efficient to make the power, true - but then they roll the cost of building and maintaining the powerplant (and the dam) into the power rates - and they aren't cheap.
Oh, and you want to kill the alternator, you have to pull the control wires - AND the Battery lead. Yes, there are some models that don't need any external excitement at all - If it's spinning and the big Output wire is connected to the battery, it's putting out ~13.8 Volts.
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