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Ian,
I bought a nice aftermarket laptop PSU at Jaycar which runs most laptops complete with multi adapters which also set up the correct voltage and polarity for the laptop. Unfortunately I can't see the one I bought in the catalog so I can't give you a link.
A bonus with that PSU is that it has a 12 V input as well as a USB charging output. That would solve your problems by allowing you to run directly from 12V.
Regarding the inverters, your problem may be the wired connection from the battery to the inverter, when running at 300W your inverter will draw around 30amps, even a fraction of an ohm resistance in the DC path will cause a voltage drop, for example at that current 1/10 of an ohm will result in a drop of 3 Volts, basically the inverter would have likely stopped. Laptop PSU's can draw up to 90W, so your current draw at 12V is still significant. If your inverter has removable leads, try measuring the voltage (under load) at both the battery and entry point to the inverter.
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