Having approached via Hannegan/Perfect Pass twice, and exiting via Wiley/Eiley twice, and in and out Access creek once, I would vote for accessing the northern Pickets via Hannegan and exiting via Access (unless you just climb in the Challenger cirque). Wiley/Eiley is pretty long and nasty but quite pretty (I won't go do it a third time, however).
Friends of mine just did a 7 day traverse entering via Little Beaver/Whatcom Pass and out Access creek (I was supposed to be there but house renos took away my annual (almost true) picket trip). They climbed Whatcom, Challenger, Luna and Fury along the way. They said getting around Whatcom was pretty easy. They were at perfect pass in an easy 1 1/2 days from the boat drop off (1/2 day of trail walking - 12 miles to camp before whatcom pass - and then an easy day getting up the pass and then around Whatcom. They had planned on the north ridge, but they were getting hammered by the snow and rain.
So if you want to avoid a long car shuttle and don't mind a boat ride in, Whatcom sounds good as the access point for the Northern Pickets.
With a 8:00 am boat ride and a light pack, one can be at Luna Pass in a day from the boat launch via Access creek, although it is a long day. You can also exit from Luna pass in a day without a boat ride. Makes a 3 day Fury/Luna trip possible.