RoyB hit it on the head...
You need a Pure Sine inverter. NOT ONE BOUGHT AT WALMART!
The inverter will say Pure Sine (PS) or True Sine. If it doesnt it a a modified sine.
PS inverters are outputting clean AC just like at your house. This is what electronics want, and what they are designed for.
You can INSTANTLY blow up expensive electronics on a modified sine inverter. There is a reason PS inverters cost more.
The additional heat is because of the voltage varaince that a modified inverter is kicking out. This is destroying your power brick. Fortunately, the power prick will burn itself up before it (usually) will output harmful voltage to the laptop. This is not always the case. I have heard of MacBooks instantly destroyed because of being plugged in to a modified inverter.
Solutions... find a 12v power adapter or buy a PS inverter.