Oceania (Australia, New Zealand & the South Pacific) - Hotels Near WLG?

I'm considering booking the nasty 6:40 a.m. NZ flight from WLG to SYD. Terrible time, but there seems to be no other option for getting to SYD by 2:00 p.m. - and connecting through AKL raises the price considerably.

Any hotels near WLG with courtesy shuttle service, or is it just taxis from the CBD that early in the morning?

And does an international Koru lounge at WLG really open two hours ahead, at 4:40 a.m.?

Just taxis.

Immigration opens approx 0430. NZ lounge opens about the same time.

Just taxis.

And the SuperShuttle, although to be honest at that time of day it should only be NZ$5 or so more to get a cab, since you'll whizz through empty streets to get to the airport.