MOSFET driver IC not supplying enough current
You are trying to go really fast!  The specified turn-off delay time is 175 ns and the turn-on delay time is 40 ns. 80 ns pulses are going to be tricky if not impossible.

You will note that the HIP4080A gives has the following spec:



From this you can see that the output voltage drop is about 1 volt when the output current is 100 mA, implying an output impedance of 10 ohms. Since the gate threshold is between 2 and 4 volts, you will only be able to pull down at between 200 and 400 mA once you get to the Miller area (where the turn-off current is needed to avoid shoot-through). You just can't push these muscular FETs around with the driver in this device. To get to the rated sink current, you would need to have the operating point at (i.e. shorted to) the 12 volt rail. I give the spec writer some credit for pointing this out in the spec, but I can see that it is still a little misleading since there is no practical application I can think of that would operate here.