You're going to be in for better than electric shock therapy when your feet hit the pavements in old Sydney town1
There have been no end of regular current affair reports on people queuing in Sydney on weekends to view rentals and then people allegedly offering higher than rental asked to get a place, so first hurdle to overcome.
Then you mention pets and you may not even get permission to queue.
I do think landlords, especially closer into Sydney can afford to be very selective and even without petted tenant experience they have probably heard of stories of destruction and/or fleas and/or annoyed non petted neighbours - barking dogs apparently being the top complaint to councils in some suburbs/cities.
And mate you're hearing this from a two dog fella but then rural life is beaut,
but if I had a city rental I'd even hesitate to rent to people having pets, having once moved into a flat full of fleas!
I'd reckon your best bet is going to be get what you can of max convenience to work/lifestyle and then work on the pet housing once you're settled as it is going to take quite a few months to organise the dogs through quarantine requirements.
If you google "Sydney Pet Friendly Real Estate", other than a heap of holiday places you do come up with
2513/101 BATHURST STREET SYDNEY if you have got a cool $650,000 but not much running room for the dogs.
I reckon you'll be better off re finding something for the dogs if you go further out and keep Sydney for visiting.