Your working holiday visa combined with your type ratings will be an issue. Most airline operators with long haul types will be after long term, full time employees considering it could take you 6 months or so to get you in to the QA system and approved. Contracting here is virtually non-existent here compared to Europe. The only ones I am aware of are Korr and Aviation Labour Group (ALG), but they tend to be more of recruitment agencies rather than true contracting companies that you may be thinking of as per the UK/Euro companies. Heston MRO (formerly AMSA) maybe be an option, but again 6 months will be an issue.
Not sure if there any issues with earnings, tax and super on a working holiday visa also bearing in mind the normal work on that type of visa would be fruit/veg picking or hospitality industry so a bit lower paid? Any issues with going from a holiday visa to a sponsored visa whilst in the country also?
You may be able to find something in the GA side of things, but it will be unlicensed as you will more than likely require a CASA Licence and types as the work would mainly be on VH registered aircraft, unless Hawker or one of the other biz jet handlers have anything?
If you are long term serious about making the move here and staying in the industry, I'd be looking at a full
PR Visa or State Sponsored visa if they are still doing them - South Australia did them for Aircraft trades but that was 10+ years ago. Good aviation jobs here even for citizens/PRs are pretty hard to come by anyway.
Sorry not to have some better news for you.