Achieving higher current at the lowside MOSFET to pulse LED
The first circuit won't work very well because the FET that receives the signal from the Texas device is in source follower mode and the source output voltage will be a couple of volts lower than the gate voltage. Also there is no pull-down resistor for when that first FET is turned off - this will likely leave the main power FET continuously turned on with its gate floating around. Eventually this may cause the power FET to become damaged due to heat.

Circuit 2 is better but you have shown two NPN transistors instead of a push-pull NPN-PNP pair. Assuming you meant NPN-PNP the drive voltage to turn the power FET on will be better than circuit 1 and this 2nd circuit will adequately turn-off the gate.

Why are you unable to drive the gate of the power FET directly from the Texas device? As far as I can tell the texas device has a push-pull output and will drive the power FET's gate directly: -



Here's a circuit of a similar device driving a motor via a single MOSFET: -



In other words, people are using it for similar applications without any extra silicon between the power FET and the Texas device.