You should get more than 600V from 3VAC, I make it nearer 850V.
If the transformer increase a 1.5V square wave to 300V, that's a turns ratio of 200, 3V has a peak voltage of 4.24V, multiplied by 200 and rectified into a capacitor this gives 848V.
Of course it's probably operated in flyback mode so you'll probably get much less than 848V, if connected to a sine wave, it'll probably be nearer 85V.
Of course you probably meant 3VDC.
I don't know whether the transformer can take 600V on it's secondary, it probably can but you'll need to double the driver frequency to prevent saturation.
I wouldn't push it much beyond 600V because the insulation probably won't stand it.