Can't help with current license/camp permit fees, as sadly I haven't been able to make a trip since some of the new requirements went into effect.

Guns, forget about it. No firearms in the Q and taking guns into Canada even for hunting trips, requires paperwork.

Get a couple of canisters of Bear Spray. Make sure you get a product that is labeled for Bears/Animals, NOT self defense. UDAP makes good ones, there are others as well. I'd get a canister for both adults.

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Dogs, plenty of people take their animals. If the dogs are well behaved that eliminates the issues on portages with animals not staying with their owners. If they bark incessantly, that could be a problem, even in the Q.

I don't know about Eagles taking your little dogs. There are plenty of them and I suppose they could if they wanted, but I'd be more concerned with a wolf or maybe lynx, deciding they would make a good snack than an eagle.
Also, PFD's are not just for people. I'd invest in a couple of doggie PFD's with the lifting handle on back in case one of the dogs decides to exit the canoe and can't climb back in.
Pretty sure the dogs shots/papers have to be up to date as well.

Adults need Passport or an Enhanced Drivers License. I think you need certified Birth certificates for the kids, but not sure.
db is right, cheaper going in from North( Canada). Might be worth the extra drive?

Way cheaper to stay in BWCA, but you pay for what you get, and until the Q loses it's luster to you, then you'll have to decide if it's worth the price.