SPL,
Sorry someone is going to need to fill me in on the rules. I haven't found anywhere where it specifies a different requirement if you're in command (ie solo) or dual.
What rule is your school worried about breaching by letting you fly solo without one, that they haven't already breached whilst letting you fly dual (from Bankstown)?
In the real world (not suggesting this is the law) - if you can access to your aircraft (ie through your school's hangar or similar), you're sorted.
Operating the aircraft without one is the easy bit - what if you flew from your school at a non-security designated country airfield, to do touch and gos at Bankstown, the return to your country airfield. Do any rules stop that occurring without an ASIC?
VI