I have done this trip before, and there really isnt going to be much problems, its a 2 hour flight at most normally, and the security at both airports for travel between isn't that bad.
There shouldn't be any problems with security about that stuff, less while wearing a diaper, its so thick that it looks like you are smuggling something... which I doubt you would wear things to make a diaper obvious, and so long as you dont take any metal through or have anything in your bag that will cause a check, they wont really check you or your stuff.
After that, its just a 2 hour flight, so even if you end up using your diaper, so long as its just wet, it should be fine till you arrive in Melbourne, where you can change in a near by bathroom (normally near the gate you enter) so long as you made sure to bring a spare diaper and changing supplys in your carry on bag.
The only thing that may be a problem, would be if you messed your diaper, which should be changed right away, and in that case, the planes used between BNE and MEL are already very small, so the bathrooms would not be ideal at all for this, so if you are also fecal incontinent, you may just want to take something to fully clear out your system the night before, so you have nothing to void during the trip, as well as cut back slightly on fiber on the day, till you are in Melbourne.