Can I drive a high-side N-channel MOSFET with NPN transistor and extra battery? (H-bridge)
While conceptually your schema would work to turn on a high side FET there are several major issues with it:


You need to ensure that the FET device you use can withstand the +/- voltage across the GS. In your case with the IRLB8748 it can tolerate +/-20V. Other devices might not be able to tolerate this high a drive voltage. 
The very high value pullup (turn on) resistor will result in very slow turn on times. In your case the FET has quite high gate capacitance and you have to charge this from 0  to about 16V to turn om the high side FET switch.
The very high base value resistor on the 2N3904 will result in long turn off times for the high side switch. With only uA of base current your transistor will act as a constant current discharge, and with high gate capacitance this takes a long time to turn off the device and could result in significant shoot through current.


Most bootstrap high side drivers can deliver several amps of charge and discharge capability for the FET gate. There is a reason this is done ….it's predictable and fast, and your schema is far from predictable or fast. 

I Would strongly recommend you do not use your schema as is without significant changes. Read up on High side drivers as a starting point. Perhaps start here, before deciding how to proceed.