I don't remember the mosquito's along Lake Superior north shore being bad that time of year. Gooseberry SP is a nice stop along the north shore, similar to the UP along Lake Superior if you like that. If you are going to the Badlands I would also stop at Custer state park, Wind Cave and or Jewel Cave also. Wind Cave has a CG and is a lot easier to get a campsite especially during the middle of the week in Aug and you can visit all the other sites I mention from there. In September as you know the kids are back in school so finding camp sites at the NP's and SP's are a lot easier. Being a school teacher you might find a trip to Mt Rushmore interesting also. Good luck on your trip.