US passport holder here. Was rejected for ETA because I disclosed my California "wet-reckless" misdemeanor (applied 19 days before my trip). Stumbled across this thread last week mid-panic. Followed the advice. Submitted 3/15. Granted 3/21. Some details:

I used Accurate Biometrics to get my FBI background check on 03/14 and received my results on 03/15. Submitted my application for subclass 600 visa that same day. I included pay stub, receipts for flights, lodging information, credit score, copy of my passport, most recent checking account statement, court document stating that I completed my probation, and a cover letter. The cover letter just had my travel dates and a brief explanation of my circumstance. The passport copy was not certified. Following the advice from the Americas Service Centre, I also had my state level background check done which arrived a few days later. I was not asked for it or any other additional documents.

I don't stress often, but the Americas Service Centre had me freaking out when they said it would take 30 days. I'm so glad that wasn't the case. Many thanks for all of your help!