First off, I love going in Aug with kids. Temps are warm, and bugs tend to be down. We started taking ours when they were toddlers (now 12,15 + 18). I've taken them by myself as well with my husband. We prefer smaller water ourselves - besides feeling safer, I think there is more to see. With regards to your questions, I'll reply to the places I've been to myself. You already know that generally you're going to encounter more people, so just get an early start and don't sweat it...you'll find a site. Try to get to a site by noonish if you can. 3o'clock is too late in my opinion.
6 - Slim Lake: While we haven't camped in this area, we did day trip there in Aug for fun. It felt very quiet and none of the campsites were occupied. The first site on the western side of Slim is very nice. I believe there were some nice rocks that looked good for swimming or lounging on. We lunched on the site at Rice Lake. It has a nice western view and is wooded. I don't remember what the tent pads were like. Rice Lake must be fairly shallow as there were lots of wild rice plants.
27 Snowbank: I found the area as well as Disappointment both busy areas. We also got caught on a very windy day on Snowbank - not very fun. It wouldn't be my first pick with kids.
41 - Brule: While a nice lake, I think I'd skip it with younger kids. The wind is unpredictable - as you know it's a big lake.
38/39: Sawbill: This is a gorgeous area with many route options, north, east or west. Yes, it's a popular area, but people are headed in all sorts of directions.
Even seeking out a site back in a bay may help you feel further from the crowds. You could head up the Kelso River and camp along that area. The portage from Sawbill into the Kelso River is a cinch. Campsite 872 on Kelso Lake is beautiful with some rocks that you could swim off on the shore. Downside is that it has been used quite a bit, but I still thought it was pretty. More of the traffic should be headed in other areas as the Kelso-Lujenida portage is pretty long and it keeps the crowds away. The Kelso River system is very scenic. You'd have options to day trip down into Alton or north into Zenith. I've been north to Long Island, Cherokee as well as east through Smoke, Burnt, etc. All of those areas are scenic with a variety of site options and day trips.
I agree with not moving camp much. We've done it both ways with our kids and find that not moving more than a couple of times seems to keep everyone happy.
Lastly, maybe you've done something like this, but whenever we bring anyone new(er) to the BW, we have a fun BW 101 day in our yard and at our local lake. Geared toward whatever age we're dealing with we cover both land and waters skills. You don't need to cover every last thing with young kids, but getting them to participate makes it fun for all and they start to get a sense of what to expect. Plus, since you're not going until Aug, you can break it down...one weekend you cover aspects of the tent (camp in the backyard), another weekend you cook over a campfire or take the canoe to a local lake (you get the idea). When we packed with young kids in the early days, we just used their school backpacks and all they carried were their clothes, camp shoes, water bottle, whistle, stuffed animal/book, and raingear. They usually hiked the portages with their pfd's on...or carabinered to their pack.
Best wishes for a really fun trip! Let us know how it goes.
Photo from when my friend and I took our kids (and 1 extra buddy) one cool August. We put in at Moose River and base camped at Agnes. By this time every kid, except the youngest, could portage the canoe or pack, and set up camp. :)
Setting up their own tent.
Feeding tweens + teens.