Consider that when this siren was new, alternators were rated to produce tens of amps, not hundreds.
The battery will provide whatever current a mechanical siren draws, even if the mechanical siren's startup current is higher than the alternator's rated output, as this current draw is only seen briefly. A mechanical siren's motor is a lot like a starter motor. The alternator will recharge the battery, just as it does after the battery provides whatever current the starter draws, as this startup current, too, is much higher than any alternator's rated output.