Try RA-Aus instead
Do you need a 4 (or more) seater?
I wonder if it is much easier to get your recreational certificate from RA-Aus. This limits you to to LSAs, but something like a Jab 230 will outperform a C172 (so long as you only need 2 seats!) and be much cheaper anyway... (At least $100/hour cheaper.) If you want something more like a Cessna, a high-wing Tecnam will do.
I would not be surprised if with a quick 5 hours, the CFI will sign you off and you are good to go.
The RA-Aus operations manual only mentions "PPL" as previous experience, not "CASA" or "Australian" PPL.
Still leaves the ASIC issue, but you can have a very good time flying in Australia while avoiding airfields where you need one. (Not to mention going into ones where you do anyway because nobody ever checks. Not that I advocate you doing this, of course!)
As a side note, I got my ASIC as a foreigner and only a provisional residency visa with very few questions asked and was returned to me within 10 days by RA-Aus.
So you might well start your tour off without one, to get one sent to you a few weeks in.