If the invertor is overheating, it sounds like you only have a 100w inverter or something.. or just a really crap one !
Higher rated inverters would run cooler as they would be using far less than what they are expected to take.
To be honest the best idea would be replacing the PS2 PSU with a specific car one like the ones posted above.
After all with the inverter you are converting 12v to 240v then converting it back to 12v, the efficiency of inverters is about 90% so that give you a 10% power innefficiency lost through heat etc..