ADG, Have you sought a second opinion on that Visa requirement for internal Private flights?
As long as you aren't flying for, and being paid by, an Australian company, I'm not sure that applies. Otherwise, anyone here for a business meeting, whilst being paid by their employer, would need such a visa and that's not the case. Its a given, that if you are here on business, you are being paid by your home-country employer.
I suspect there is little difference to entering Australia on a CTA for Commercial flights/Business visa for Private flights and entering the USA on a C1/D or B1/B2 in the same situations.
But then again, I don't work for Border Force. I'd ask your Handler to check and get it in writing.
Best of luck.