adding a buzzer would be very annoying, and it's simply unnecessary. you shouldn't have to be reminded to turn it off, you should have it wired to turn on/off automatically. this is easy to do with a relay. make sure you use a 12VDC relay with an amperage rating higher than your inverter's maximum current draw. if it's a standard bosch-style relay (the most common automotive type relay), then you should wire it like this (your relay may be different, so always check the relay's instructions for proper wiring):
in your case, the wire labeled "to carputer and accessories" should be input for your inverter. for the relay coil input (usually lead #86), you can use any switched 12vdc power source. you can use the car's ACC circuit or the car's IGN circuit (they are not the same);
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the IGN circuit is powered when the key is in the 'RUN' and 'START' positions, but is NOT powered when the key is in the 'ACC' position. power will stay on while the engine is cranking. however, in some cars, leaving the key in the 'RUN' position for extended periods of time with the engine off can burn out the ignition coil (I've never heard of that happening to anyone, but that's what we were taught in high school automotive class/shop

), so if you plan to use the carputer with the engine off then I would recommend wiring it to the ACC circuit instead.
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the ACC circuit is powered when the key is in the 'ACC' or 'RUN' positions, but is NOT powered when the key is in the 'START' position. this is good because you can run the inverter/carputer without the engine running (and without the risk of burning the ignition coil out). the downside is that power will turn on briefly while you turn the key, then turn off while the engine is cranking, then turn back on in the final 'RUN' position.
good luck