Is there a possibility that they've assumed you've landed because you did the ILS and not considered the fact you carried out a missed approach? If they knew what actually happened they would have sent the account?
While their charges for GA are daylight robbery (Dick Turpin would be proud of them) perhaps they should be given the benefit of the doubt meantime.
I don't see how for the life of me any airport could justify a charge in these circumstances. As I see it Cairns Airport didn't provide any service. Also it's not like this aircraft, by flying the ILS, deprived them of the opportunity to derive income from another aircraft.
I wouldn't be paying, that's for sure, but before I made too much of a fuss I'd contact them suggesting perhaps there was some sort of mistake.