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Mine is fused at 10A (it says so on the front), which is only 120W - but as long as you're not drawing too much power from your inverter you 'might' be OK. Try it and see - what are you trying to power from it?.
To get the full power you could either put croc clips on the inverter and connect it direct to the battery via a fuse of course, or if you need to use the inverter while moving put another socket in the car capable of supplying it.
The efficiency of the inverter and the total load will determine how much power you can pull but typically a automotive cigarette lighter or power port socket is good for about 100 usable watts at the end given they typically have a 10 amp fuse and that type of socket and plug system is not designed to handle more than that in good working conditions.
I know of many who put larger fuses in so that they could get more power only to end up with burned out sockets or melted plugs for their efforts.
As for powering an unloaded inverter I have ran 750 - 1000 watt rated inverters off of such power ports many times, given the actual inverters idle currents were quite low, so your 300 watt unit will likely run just fine until your load passes whatever limit the fuse sets for the system.
The fuse amperage of different car cigarette lighter is different. You first of all should know your car's cigarette lighter fuse. My car is 15A. I have a 300w power inverter bestek for my car. I often charge cellphones, tablets, cameras on the road. The car cigarette lighter can load max 150W. Don't plug devices over 150W directly into the car cigarette lighter. If you want to use it, you can connect the two clips with car battery.