Several things to think about (with my ADI hat on).
As others have said, just getting to the biting point with the clutch should be enough to release the handbrake automatically with that sort of car. BUT... the stupid brake mechanism in all the ones like this I have driven are too slow to release, so if you try and move too quickly you will add too much load onto the engine before the brake releases and you will stall. A manual handbrake is so much better, but that's another thread altogether.
Most importantly though - why are you thinking about hill starts as something special? They aren't. All starts in a manual (except when facing downhill, when there are options, but a normal start will still work fine) should be the same - there's nothing special about a hill start. Each and every time you have stopped long enough to get the handbrake on, you should find the bite so that the front of the car rises slightly, then release the handbrake (through whatever mechanism) or allow it to release automatically where available.
Annoys the crap out of me when instructors build it up as something special. Just makes people nervous of hills when they should just be doing the same as they do on the flat.
Second thing to have in your mind is that your gas/clutch sequencing needs a little change. You seem to have been taught to simultaneously add gas and raise the clutch (many people are), but this isn't quite ideal and can easily lead to a stalling problem.
Always, always, always lead with the gas. So add a little gas, then move clutch to the bite. Release the parking brake. Now being to gently add more gas. As you hear the engine accelerating, slowly start to raise the clutch.
The reason is that bringing the clutch up through the biting point is adding load to the engine. That slows the engine down as it's working harder. If it slows too much you will stall. So always, always, always speed the engine up first before adding load to the engine. That way, you should find it almost impossible to stall.
Good luck with the lesson...