Why is the MOSFET driver in this circuit dying?
One obvious difference between a resistive load and a battery is that on startup, the resistive load is zero and the battery obviously isn't - it's a source after all. This is called a "precharged load" and can be messy without some form of active or passive OR-ing control. 

You don't have any isolation between your synchronous buck and the battery, so if your PWM commands low duty cycle, your "T2" (it really should be Q2 or U2, since T is generally used for transformers) will be on with a wide duty cycle and sink current through the inductor. Your synchronous buck may start acting like a synchronous boost, or even saturate out and short the battery. As "Some Hardware Guy" stated, when the battery is connected you may see some interesting things on pin 6. 

Also, there are some intelligent buck controllers out there that can sense the low-side switch drop and shut things off if heavy current is detected. The IR2104 is a high-side / low-side half bridge driver, which can be used to drive a buck, sure, but it's not the best choice.

Try a big diode between your buck and battery, and see if the behaviour changes.