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You'll find BWCA permits for Moose Lake available just about any day in the summer except most Sundays and Mondays.  As IC posted, reserve an overnight BWCA permit for the same day you're entering Quetico.  Pick up your BWCA permit at the Kawishiwi ranger station (or an outfitter if you're towing up to PP), and then hit the water.  Get your Q permit at PP, paddle the Q portions of Basswood, then head to the US side for as many days as you want.  You will see motor traffic on parts of US side, although motors aren't allowed north of Washington Island to Basswood falls and areas just to the south and west of the falls.  Fishing is good almost anywhere in Basswood.  IN addition to your permits, you'll need Ontario and MN fishing licenses, ON Outdoors card, your RABC, and passport.  When you finally exit back to your vehicle, you just need to visit Customs in Ely.