Unfortunately, out of "sturdy, light and cheap," you can generally have only two. Sometimes only one….
In my personal humble opinion, the best bargain for a light travel tripod that will hold up a 200/2.8 is the Feisol traveler, available here:
http://reallybigcameras.com/Feisol/Traveler_Class.htm
The 3441-S weighs a kilo, folds to 43 cm, opens to 128 cm without the center column and 178 with, and costs $300. The 3441-T is 16cm taller without the center column raised, which is better for taller photographers. Either will fit in a carry-on bag on an airplane, and either should easily support a 200mm lens with the proper ball head.
The same company sells the Korean made Photo Clam ball heads. I have not tried one, but the reviews online are generally positive. I am planning to purchase the 33mm or 36mm ball head for my travel tripod. They run less than $200.
To solve the problem you are having with the horizontal-vertical transition, an L-plate for your camera is very useful. You can get one from Really Right Stuff. Also, if you are going to work on a tripod, you can safely remove the battery grip. It adds one more screw-connection that can flex, and it's not all that useful on a tripod anyway.
These prices are not all that bad for good photographic equipment. My main "work" tripod cost well north of $1500 with the head and the proper camera and lens plates, but then I use it every day and it was deductible. The article on the bythom website does a very good job of explaining the benefits of buying your last tripod first. (And I am a poster child for the long line of mediocre tripods and heads purchased over a twenty year period before finally buying a good tripod.)