Do you have a similar 'permit to fly' system for experimental aircraft there?
Yep - experimental amateur built aircraft are massive in Australia. I have one.
They are issued a Special Certificate of Airworthiness:
https://www.casa.gov.au/experimental-certificates
CASA recognises UK:
https://www.casa.gov.au/sites/defaul...agreements.pdf
Certification of Amateur Built Aircraft (Part 13 building outside AU):
https://www.casa.gov.au/certificatio...-abaa-aircraft
More on commercial assistance for amateur built aircraft (basically engines/rotors/avionics excluded, but fabrication & assembly required):
https://www.casa.gov.au/commercial-a...built-aircraft