The night before a public holiday you probably wont get a mooring at the Basin if arriving late, but you can easily anchor just outside the mooring area, it wont be good in a NE'r tho' anyway, the better option would be Refuge/America's Bay, I think America's is quieter, better protected, & you get to sleep in as the sun takes a bit longer to reach the boat from over the hill

, there are hundreds of moorings there, so you should be ok.
I have found the southern end of Pittwater a real nuisance unless you have a mooring/berth, years ago I got away with anchoring in Winneremery Bay while provisioning in Mona Vale, but wouldn't attempt it nowadays. I reckon you'd be heaps better off stocking up before you left LM, & just pick up some fresh stuff from Cottage Pt or Akuna Bay as suggested above, by the time you factor in bus/taxis cost would be about the same....without the hassle. cheers