Essential flight crew only means just that - only the essential crew may be on board. No passengers, students or whatever. I'm not familar with a Mooney Ovation per se, but if it can be flown, according to the regs, by a single pilot, then "essential crew only" means single pilot. Period.
If the regs says that it is a multi-pilot aircraft (like the Boeings and Airbuses are), then you need two fully qualified pilots on board. If you're flight testing, then the essential testing crew would obviously be allowed on board too. Still no passengers or student pilots.
If the flight is long and there are rules regarding duty time for this particular operation, then I would guess that two (fully qualified) pilots could be carried, one being the relief for the other. Still no student pilots or passengers though.
Furthermore - if the CFI is not registered and/or authorized to conduct training in the US, then the hours that he lets the student pilot fly cannot be considered flight training. For all practical purposes the CFI is just PIC and the student pilot is, well, a passenger along for the ride. So these hours cannot be written into the student pilots logbook and thus cannot be counted towards any experience requirement.
If the student were to fly the aircraft solo, it would be even more interesting. The CFI is not allowed to sign the student off to do solos, the student does not have an FAA class 3 medical (which is the student pilots license as well), the student does not have TSA clearance (required for flight training in the US in any case) and I doubt whether the insurance will cover "flight training" at this stage. I would also expect that the Flight Permit (both AUS and FAA) will specify that only flights may be undertaken for the specific purpose of delivering the aircraft to Australia.
The fact that the CFI has a US license (a CPL or ATPL I guess) doesn't matter one bit,
AFAIK. The aircraft is Australian-registered so in principle you need an Australian pilots license to fly it around the world. The fact that the FAA has granted a special permission to fly the aircraft in the US is probably because the Australians have yet to issue a regular CofA. Only aircraft which have an ICAO-compliant CofA will be admitted into or flown in other countries airspace without further paperwork.
I'm not an aviation lawyer specializing in FAA regulations, but I don't think you'll ever going to make legal...