It is unfortunate that OAA accommodation has been discontinued.
I have had a number of families that have moved to my homes after a short period at OAA. The last family described it as...well, I can't possibly put their comments on here but their best desciption was "prisoner in cell block H"!
I moved here 4 years ago and stayed in short term accommodation. I thought that it was expensive until i moved into my own house and the reality of the bills kicked in.
One thing that you really need to take into consideration is that there are some areas that you would not like to rent in. They are cheaper...but they can be "dodgy". It is so important to start out your new life in the best possible way. You have already ticked every stress box there is and if you make life difficult for yourself right from the beginning saving a couple of hundred Dollars...my opinion is that this is a mistake. You will soon find out that Australia is very expensive and the initial cost of establishing yourself will soon blow out well beyond any 4000 pounds.
Most families try to substitute their established lifestyle in the UK and not take a step back hence a good rental in a good area, with all the things that they had in the Uk.
If that's not your choice so be it but it is again my opinion that you then make it harder for the family to accept their new lives if they have taken steps backwards in terms of creature comforts.
West Beach challet and caravan park have tent sites at $32 per night plus $20 per child per night. If there are 4 of you that would be $72 per night to stay in a tent! I rent my fully furnshed cottage with everything for $75 per night....and this is why I don't consider furnished rentals to be expensive.