A TRI qualification is of no use unless you are operating through a Part 142 organisation. The 142 organisation will be able to assist in converting your TRI/TRE to an Australian equivalent. I also suggest you contact Tony Stanton and put your situation to him. Tony is heading the CASA Part 61 Task Force whose aim is to tidy up anomalies created by the new reg's. Given the whole rewrite of FCL reg's was to harmonise with ICAO, CASA should accept other ICAO signatory state qual's. Also have a look at the CASA Licencing procedures manual, Page 105. Below is a link to the manual.
https://www.casa.gov.au/manuals-and-...cedures-manual