With a towing rating of 8200 lbs, you have more truck than needed to tow a 2200 lb dry weight camper by a factor of at least 2. You won't even notice it's back there, try to not fall over from shock when you arrive and realize somebody tagged on a trailer while you were traveling.
You got nothing to worry about.
Oh, the reason it wants to downshift at every change in grade is that it is set up to squeeze every inch of travel out of every drop of gas. While you are towing, it will certainly get poorer mileage than when you are not towing, so expect it to use lower gears than normal pretty much all the time. But that's only because of the programming. In the old days, they just would not have given you those extra tall gears, the ones needed to squeeze every drop of fuel 'till it screamed for mercy.