Well, I think that $1500 is super tight for a week and a family of four. I really love Colorado, and it is just a beautiful place to be in the summer. If I were in your shoes, I might consider flying into Denver, renting a car, and heading over to the Keystone/Vail area for the week. Vail is pricey for condos, but you can find a very nice 2 bedroom condo in Keystone for $99-$130/night, via VRBO, which will put you at about $910 in lodging costs. An intermediate size rental car will set you back about another $260, bringing you to just under $1200. The resorts at Keystone all have nice outdoor pools and lots to do for kids. You would have a bit of play money left for a few activities. You can explore Vail and Dillon (both cute towns), hike and bike for pretty much nothing (you can rent bikes at Keystone Village). Many condos have a good supply of DVDs and games, but you could bring some from home to occupy your evenings. Vail has a really nice outdoor amphitheater and they have free concerts every Tuesday evening at 6:30 pm. I would just take your weekly food budget and put it toward food in the condo for the week, as well as your weekly gas budget to help fill your rental car, and would figure those costs in addition to the $1500.
That free airfare is enticing, but the problem is that you almost always will have to incur the cost of renting a car if you fly somewhere which cuts into your limited budget. I live in MN too, and I think if I were in your shoes, I might just drive over to Door County for the week and save your air travel for another time, as I think your money will get you further. BTW, I love the North Shore too, but I totally get your wanting to try something new.