Extending a switch mode power supply controller IC to work under higher input voltages
No this won't work for a few reasons. Firstly the simple reasons: -



Original answer #1 was incorrect so deleted and answer 2 edited


First answer deleted
Driving the FET with the collector of a BJT is not good for the speed of the pulses. The collector resistor has to discharge gate capacitance and this will cause gross inefficiences. 
The gate-source region will be destroyed by too much voltage due to the BJT pulling it down to ground.


The more complex reason: -


Even if you could cure the problems above there is the issue of loop
stability which I'm afraid will kill-off this idea completely. You
now have introduced two gain stages (cascaded) into the feedback
system of the basic chip and this will never stabilize correctly -
it'll sing and ring and oscillate.



  what is the correct way of doing it?


Recommendation - keep the chip in the draw and find a proper switcher that can do the job.