Have you decided where to stay yet
Be ready for an early appearance at EK's zone check in at MEL T2 tho.
QFd ADL might not issue an EK BP, even on QF's BP paper stock, as its overnight stop, they might if it was same day, but overnight, you would probably have to start from scratch in MEL at 3.30am/4am at the EK counter.
On google, it lists EK409 dep at 6.30am, so check in would probably be 3.30am to 4am.
Give it 15mins to half an hour from swing open first BP scan to get into secure area, then passport DIY, then int security screening, then walk all the way to gate 10 upstairs, might be 4.45am, if you start the sojourn at 4am.
I had done it once with EK F MEL - AKL a long time ago, pre covid when EK used to fly east coast Au - NZ.
The F lounge is definitely better than the QFi MEL J lounge in the basement!