I've done several fly, drive, and camp roadtrips and I've always left the stove at home. The problem is the fuel, and not the stove itself, but if the stove uses any kind of liquid fuel, it will almost certainly leave a residue that will be flagged by security. Propane stoves might be ok, since the fuel is a gas, but I've never risked it.

I think the fly one way is a great idea with the time you have. However, you might consider reversing things, flying there and then driving home. That could make the packing issue much easier. You'd be able to figure out what you can fit onto the plane from the comfort of your home, and anything extra you pick up along the way, you'd only have to worry about fitting into the rental car to bring home.