DC-DC Boost converter IC with external MOSFET?
I very much doubt that this will work at all. If you look at the block diagram for the FAN5331, you will see that it senses the current flowing to GND through SW. It uses this current as an input to the feedback loop which maintains regulation. If this current is not proportional to inductor current, I think the output regulation will totally fail.

Also, while I did not scrutinize your schematic carefully, you would certainly need a resistor in series with the base of Q8. Otherwise you are going to blow up Q8 by putting 5V across the emitter-base junction. Whenever you use a BJT as a switch, you will need a resistor in series with the base to limit base current.