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.