Unfortunately no.
The only alternative you have is to register either yourself or a company as a business (get an ABN) and then register for GST. Then you can claim back the GST paid on any purchase (including maintenance, fuel etc)
But, you would then need to invoice yourself for flying the plane (at a market rate) and include GST in that rate. Whilst no money would change hands, each month you'd have to send the government a cheque for the GST 'collected'. And when you sell, you need to include GST on sale and return to the ATO (unless you're selling overseas).
It can be worthwhile - but you need to be prepared for some sticky questions should the ATO audit you. It's perfectly ok to fly PVT under this arrangement as you are not chartering the aircraft. If anyone asks, you are hiring the aircraft from yourself (rather than a flying school etc) and flying it.
UTR (not a lawyer or accountant)