A couple of years ago we cruised around Broken Bay and further abroard on a Supersonic 27, it worked well for just the two of us, and often we also did 'campout' cruises with extras onboard.
Could the budget extend at all??? As I am sure you are aware your budget is very hopeful. Anything at this price point would have to be a severe compromise of one kind or another, ie people sleeping in the cockpit......sleeping bags on the floor etc. In terms of introducing a family to cruising, i would recommend doing everything you can to make it a enjoyable experience.
If you could squeeze the budget even a touch I would highly recommend looking at a Columbia 34.
An early American design, however the aussie boats were built here and are alot newer. They are huge for a 34 footer, and can sleep 7 with everyone in their very own bunk.You would get one for 40K.
Have a look at this one. Been on the market for over a year.
COLUMBIA 34 SLOOP boat details - BoatPoint Australia
I have sailed on a Carter 33, they are a fast boat, would be fun daysailing around the lake, but smallish cockpit( I completely agree with TDW a liveable cockpit is important), feels cramped down below as well, particularly compared to the Columbia 34 mentioned above, it is a very practical layout at least. I know a bloke who cruised the eastcoast with a couple of additional crew on a Carter 33 and he made it work.
I tend to think if you get a 27 something you will either love cruising and have outgrown the boat almost immediately, or it could put your family off the experience .
My two cents and this is going to be controversial, but if you want to go outside Lake Macquarie and cruise the NSW coast(and up to QLD- I WILL convince you to do that cruise with your family) then forget a trailer sailor, the NSW coast is not where I would want to be in a TS. Just my thoughts, others will disagree.