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An alternator can't output full power when the engine is idling. You may only get a few hundred watts before the battery starts draining. You can get a generator that will produce much more power than a typical car alternator for under $400.
You're speaking like an alternator is a generator, and doesn't come in various outputs. Your average larger vehicle can charge itself from near-dead at idle or run a 3500 watt inverter, although I have had the odd small car or motorcycle incapable of either.