First of all, welcome to the forum!
I have been to the BWCA a number of times, but never used an outfitter. Unless one has an experienced person to help on the float, I would start by making calls to the various outfitters for advice and prices. They would have the latest and greatest. Expect to pay a bit more for the service. Figure in cost of food, maps, etc, all the things the outfitter recommends and just used an Excel type spreadsheet to figure out expenses. Price of gas at the time, be generous, travel during tourist season is always an ify factor, but expect to pay more than what's on the pumps now. Contact the local scout troops/council in the area to get the prices on that and use Google maps to figure out distances.
Basically , start asking questions, lots of questions. Keep it in mind that permits are limited for each entry point and you will need to have a permit for each group of nine people. These groups cannot travel together, nor can they camp together in the same site. 9 people to each site is maximum.
Not only should the boys have First Class, I would recommend First Aid, Swimming, and Canoeing MB's under their belt. You'll need a Wilderness First Aid person with each group of 9.
Hearsay of those doing Alaska (my wife used to live in Alaska) is not very good. One of our troops in the area contacted an Alaskan Council to take the boys up there, but they were not very helpful and the troop didn't go. If you know of someone who knows the area, that will be a big help. Unless one is flying in or taking the inland ferry, you will also need to know Canadian/US border crossing issues and each boy will need a passport. Gas is a lot more expensive in Canada and the exchange rate changes on a daily basis so planning that far out could pose a problem for judging fuel expense until the last minute.
If one has not had experience doing high adventure, it might bode well to try a lesser strenuous trek. Isle Royale is nice as is the Appalachian Trail. There are a number of less frequented trails as well such as the Centennial Trail and the Ice Age trail. For the boys who want a high adventure/summer camp experience, there's Wyoming's BSA Camp Buffalo Bill located 6 miles outside of Yellowstone National Park's East Entrance. Very nice, but get your reservations in early the camp fills up fast!
Decide what kind of trek/float you wish to do then start making contact with the locals with all your questions. They will be the best ones with the latest and greatest information. One may wish to contact a local troop (Be a Scout website will ID them for you) and propose an exchange. They host your troop in say Alaska and in return, your troop hosts them for a high adventure down here in the lower 48.