Have you considered flying with SIA?
They do SYD-SIN direct, then Singapore-Rome direct, the crew are usually very good with kids of all ages and they are likely to be more helpful about rest facilities in SIN too! although you can get a flight from Sydney to Singapore that pretty much dovetails with the Singapore - Rome flight, return flight the same.
As EWL says you are entitled to a full refund, tell your travel agent to re-book you and the funds are with QF, (if they have actually paid QF yet!), all you want is your new tickets please.
Worth a try.
Just an example from the SQ website,
Depart Syd daily except Sunday at 1700.
Arr Singapore 2155. (SQ222)
Depart Singapore Tu/Fri and Sun 0100
Arr Rome 0715. (SQ340).