I think your probably right on the money with the 243BHS. Personally I wouldn't go much more than that. The 252 puts you right back up to almost 30 ft in trailer.
I have a 2007 tahoe with the 5.3l and 3.73 gears. Pretty similar to your truck, even though you have the 3.42 gears, you also have the 6 spd transmission (I think its a 6 spd), so my thought is its probably going to tow about the same, up and down grades.
We tow a 2300 bh passport with ours, and it tows just fine with a standard wdh. Could it tow better with a better hitch? Probably. But for now it works. Granted I have not towed up and down many extremely steep grades, but for what I have done, I would be upgrading the truck prior to getting a larger trailer. A lot of it has to do with payload, and if its just you and the wife and gear thats one thing, but you add in the dog, the 2 kids, a friend...eats up your payload quickly.
That said, I think what I would do is get your payload numbers, get the weight of whatever hitch you plan on using, the weight of whomever will be in the car, and start adding everything up. Use the 243bhs as the trailer you might purchase. Remember, the numbers you will get from the dealer will likely be quite a bit lower than the weight when you add in the battery, propane tanks, and other accessories that come with the trailer.
A earlier post talks about a hensley arrow hitch. I would say thats probably one of the top of the line units out there. Find out what that costs, and see if that changes your calculation on what you might purchase.
I didn't consider this when I got ours, and I regret it. I don't think I need a hensley, but for what I want I wish I had the money back on the hitch I ended up with to get a different one. Just speaking from experience here. I also overthought the amount my first vehicle could safely tow, and after my first long trip (about 300 miles) I bought a bigger truck. My early trips were only 25 miles at max and the experience was much different.
Live and learn I guess. In the end I am happier with my new vehicle (even if I didn't have the trailer). But there aren't always happy endings to these kind of storys.