I really feel for you!
We had the exact same situation as you two years ago when we were looking for buying a property. We both work as a full time professional but our deposit wasn't enough to choose from properties in the decent suburb with a small backyard for our pets.
Also, we don't have any heritage properties from our parents who live overseas.
In result, we bought a small brand new townhouse in Bankstown. Originally, we wanted to buy out place in Revesby address at least but we didn't have a chance. But our townhouse locates close to Revesby side of Bankstown. Our new neighbours have similar stories.
Your question “Auburn vs Bankstown”. Definitely, we prefer " Bankstown". We would recommend you to look for your place in Revesby and Padstow side. If you come around this area, you can see the ethnic demographic diversity cleary.
Actually, Bankstown is mixed with Vietnamese and Arab communities and Auburn is got a strong Arab community. And Revesby area is mixed of all nationalities in good balance.
Revesby and Bankstown both have convenience fast train lines if you go to the city for work.
And you can find a lot of cheap eats and good fresh grocery shops in Vietnamese size in Bankstown.
My husband uses a train from Bankstown station to go to the city but my daughter and I use Revesby station for a little peace of mind and convenience.
Hope it helps!