The Dragonfly is a good option in terms of weight - it is indeed minimal. The carriage has its limitations though, with no suspension other than the air in the tires. What you are experiencing with your bike is called "bump steer" - this happens with fifth wheel vehicles as well, if one front wheel hits something, it whips the pole sideways. That is simply a physics thing - the wider the wheel track, the more it will whip sideways. On really rough terrain, the best hope for horse and driver is independent suspension. The heavier weights of many four wheelers is not offset by there being no weight in the shaft loops. Weight is weight - ease of pulling has many elements in it - so I would suggest any two wheeled vehicles that have independent suspension for what you want to do, and where you drive.
A vehicles with two wheels and no roof that was pulled by horses in races and battles in ancient times. Someone who drove a chariot was called a charioteer. Usually private four wheels vehicle, drawn by two or more horses. If the seat back is moving it means the front of the cart must be going up and down.