Yes this makes a lot of difference, as it rules out a 417.
As you surmised the student visa will be easier to come by then the TSS. There would be nothing stopping you transitioning into a TSS at a later date when you find a job. Just remember you will still be liable for any associated course fees dependant on what contract you sign with your education provider.
Living together for 12 months doesn't constitute the requirement for a defacto visa. You need to prove you've been in a defacto relationship for 12 months, there is no requirement to have physically resided together for those 12 months.
Also just remember whatever visa you come in on you need to be a genuine entrant on that visa. There is no visa designed as a stepping stone visa for an 820/801. (Technicality PMV but irrelevant based on what you've said).
Finally just to avoid confusion down the line, a bridging visa with full work rights doesn't come into effect when you apply for your 820, it comes into effect when your current visa expires. So if you apply for a 4 year student visa you will be bound by its conditions for 4 years or until the 820 is granted, whichever comes first.
