Hi again,
I really dont think the LM358 is the real problem here, the problem is the low voltage power supply. It wont drive a standard MOSFET properly.
As Ron was nice enough to point out too, the 33k resistor for the buzzer really does seem kinda high. Also, we dont know what kind of buzzer this is. Is it really a self contained buzzer (with internal oscillator) or is it just a piezo? piezo would require an external oscillator...just thought i would mention that just in case it is. DC operated buzzers most likely take more current that a 33k resistor would allow, and so you should check the required drive voltage for the buzzer if it isnt really a piezo. A DC operated buzzer could take anywhere from 10ma to 100ma or even higher at a specified voltage like 5v, 6v, 10v, 12v, etc. We have to know the spec's of the buzzer to recommend something more precise.