Airside international has a Cafe Vue (of Vue de Monde fame) which is pretty good for an airport but it may be closed by your departure time. There is also a bar with good snacks, fwiw. If you had booked the QF codeshare on FJ then there is access to the QF J lounge (at least for QFF status, haven't checked BA Silver) but QF codeshare often costs quite a bit more than the FJ fare on same metal. And the J lounge is hardly worth an effort, unlike the F lounge which is stellar at MEL.