That is a VERY long connection imo. There is usually an earlier OOL flight (QF880) from MEL that you may be able to move to (unless for some reason it is not operating on the day of your flights, or is already full in J - which is very possible possibly).
I would agree the domestic Qantas Club is not the best place to "relax" for hours and hours. Specially in the peak morning hours between say 7 and 9 given a lot of people wanting coffee, breakfast etc. Noting if your QF OOL flight is in J then of course you can use the Business Lounge (and actually if your incoming international QF36 is in J you should have access too) and the J side is indeed a bit bigger but still I would not want to spend hours and hours there tbh.
It's probably not worth a day room somewhere close as it's just not quite long enough in my view to be worth it unfortunately. After such a long overnight flight from SIN the last thing I'd want to be doing is trying to kill another 6ish hours in domestic. ugh.
Is there any way you could, for example, look to moving to SIN-SYD-OOL or something else or to see about changing to the earlier OOL flight? I know you have kids and that complicates things, but this is really not an ideal layover imo.
Actually thinking of the kids, who may get little sleep on the QF36 perhaps it WOULD be worth a day room at the nearby parkroyal or somewhere - expensive yes, but to get say 4 or 5 hours of sleep could be invaluable for EVERYONE. I would urge you to explore this as an option imo.