What you are seeing is voltage drop due to load. From the sounds of it, your batteries are really old or fried.
You should easily be able to charge a laptop with a 100W inverter and a car phone plug. The plugs are typically rated for 108W or about 8 amps. We have run a TV and satellite receiver through one (total 90W) with no problems but, our batteries were less than 5 years old.
Just an FYI: charging a laptop is less of a draw than using a laptop. Also, the biggest draw is to use a laptop when the batteries are low: the inverter powers both the battery charger and the graphics, which can be over 85W total.
