When we fly (usually every other year), we usually fly on SouthWest (bags fly free!) out of BNA to Denver, and each of us check two large duffles with everything except perishable food and fuel canisters. This past year we flew out of TYS and had to pay the bag fees, but it was less than what we would have payed to ship it (although it was close)...Our 4 person tent weighs about 15lbs, so its not too much of a problem to pack. You just need to make sure each of your bags weigh 50lbs or less.
However, if I was flying with smaller kids (like you), I would ship it all rather than deal with the hassle of checking it at the airport. The costs difference would be easily made up by not dealing with the hassle...
Its also important to stay organized, there is nothing worse than forgetting something when you are out in the woods and have no easy way to get the items you needed...I have a spreadsheet checklist of all of our essentials that we pack as well as things we will buy at WalMart/SuperTarget when we get to Denver. I can send you a copy if you'd like.
"Even a fish wouldn't get into trouble if he kept his mouth shut."