One option I thought of that'll hopefully get some ideas flowing for you was a Kawishiwi to Kawishiwi trip. Lake, entry point 37 to River, entry point 30. (Entry point 30 is technically Lake One, but its on the Kawishiwi river. As for the 6 day trip, you'd have to check the full distance and decide how much you want to move vs. spend time fishing ect... That's obviously completely up to you, but even with doing a out and back trip from either entry point, I think you can get some decent solitude, and likely really good fishing along the way.
Be aware that parts of both areas were involved in the Pagami Creek fire about 5 years ago, I have not been to any of those areas since, so I can't comment on the current state of the re-growth, but its typically quite slow.
Moving from Kawishiwi Lake (entry point 30) though some of the better fishing areas I've seen are as follows. Malberg Lake has always been good overall for walleye, though I never hit on any one specific spot, I've caught em throughout. Additionally north of Polly, the "crowds" really thin out. North and slightly west of Malberg is Beaver Lake and Adams. Campsites 1967 on Beaver (great views and a good jumping rock on the west side) and site 2015 on Adams are both excellent. On Beaver lake we caught both walleye and smallmouth around the small islands specifically on the east end of the lake. Northerns are of course everywhere and will bite on just about anything. On Adams I did well in both the North and Southeastern areas around the islands and the channels in between. Just past Adams is Boulder which was always outstanding for Pike, lots of submerged weed beds, and good sized walleye, never big numbers but good size. Also in Boulder the island camp in the middle was a 5-star for sure, with really good cedar stands, though I've heard that it suffered some blowdown damage some years back.
Following the river SW from Malberg you get to Fishdance Lake (check out the Pictographs) and Alice. At the bottom of the two portages between those two lakes, try for walleyes with diving crankbaits or jigs. I've heard that Alice is a outstanding walleye lake, but it's just too big and open for my taste. After that you get to Insula which is probably my favorite fishing lake in all of the BWCA.... One spot in particular which I heard someone else refer to once as "the meat house" is NW of camp 1337 (essentially the second NW finger coming from the north) The bay that butts into the back side of campsite 1337 is pretty cool to fish.... If you look at the Sat map, you'll see 5/6 little rocky islands. Fish between those islands, in the morning, while trolling 5 1/4 inch original Rapalas. Specifically in the Perch or Firetiger color patterns. I cannot emphasize the time and colors enough, not a clue why it works, but it did over 4 different trips. Other than that little back bay, the whole large finger of that lake, again following to the NW, is chock full of walleye. I've done very well trolling in the area, 15-25 feet down and also jigging around the island "between" campsites 1338 and 1339. The latter has a really nice sand beach btw...
All of Insula is good for fishing though, I really don't think you'd be disappointed. Past Insula you head down through Hudson, Four, Three, Two, One, then onto the Kawishiwi river proper to Entry point 30. While I've caught fish all along that area, Lake One for instance is supposed to hold some really big northern, the only other place I'd mention is the head of the rapids that connect Lakes Two and One, note that this is not where the portage is, this is actually the main water flow, which you do not want to try to run. But at the top, there is a small island you can jig for walleye/Smallies off of. I've always done it with live leaches, but I suspect that if you're any good at knowing how to jig, you'd be fine.
Last thought on fishing, I cannot recommend highly enough getting the proper equipment to troll. By that I mean good rod holders, I think tite-lok is the brand I have that can clamp onto your canoe. they have a C-shape clamp that can attach to any almuinum canoe with no problem or to a kevlar canoe with a couple of 3x3 plywood blocks on each side between the clamp and the canoe. I've caught a hundred times more fish just trolling along than any other way. You paddle a bit slower, but if you don't mind that, why not fish as you go. Hook on your favorite Shad Rap, Wally Diver, Smithwick Rouge, ect... I've been to BWCA/Quetico 20+ times and my parents have been going since 1971 I think.... Trust me, it works.
The one thing I should mention in all this is that most of my experience has been when going later in the year, August through early Oct.... (less bugs). So while the places may change slightly, the fishing should actually be in your favor going in June.
Most of all, enjoy the most beautiful place in North America.
Happy paddling