I appreciate this is a hypothetical question, and the OP is asking from a academic perspective. It is though a common scenario people face. Personally I wish more people would consider the horse (as per Alisons post) prior to the assumption that its being a 'brat'.
Having been in the military I know what its like to carry weight on my back and try to balance. Even with a relatively slight increase in weight. Try wearing a back pack and then running in a 10 foot circle...... awkward isn't it.
I am not saying that there cannot be issues or refusal, however, one of the problems the horse faces is ignorance. Ignorance of biomechanics and anatomy. Its much easier to assume the horse it trying to 'get one over' than it is to understand what the horse is saying. Too many people run a horse round a pen, saddle it, and have no consideration for its development or capabilities. I see many people ride and then comment on how stiff or sore they feel afterwards. Well if you are stiff when you've done nothing, how is the horse after carrying you?
On that note there are a number of factors I would pose for consideration. The first is per Alisons post. Flexion cannot be forced. I will repeat it time and again, strapping a horses head round is like me asking you to touch your toes. When you stop I will force you to bend further then tie your head to your feet. Ridiculous? Painful? Sure it is. So why do it to the horse? It wont make them anymore flexible.
Secondly balance is dynamic, when you ask for a turn a horse has to rebalance and allow for the change in centre of gravity. Next time the horse refuses ask yourself - when did I ask for the turn? Did you even think about it? I see so many people just pull on the reins for a turn with no thought for where the feet are. If you ask for a right turn ask as the feet are about to lift, then you can control the placement and the horse cannot refuse. However ask for a right turn when the feet have placed and you force the opposit leg to cross over, which is like us tripping up. This is often why a horse refuses. THIS is making the right thing easy and the wrong thing hard.....NOT working a horse in punishment in the hope it wont challenge again. If you don't know there the horses feet are then learn. Its the difference between harmony and fighting. A lot of problems will just fade away. If you don't have the patience to learn to feel footfall then you don't have the patience to be a good horseperson.
Finally as for direction its not about winning. Winning may get you a light horse through endless drills (i say many because when it doesnt the horse gets the blame as untrainable, wrong attitude etc and gets moved on for an easier prospect.). It wont get you a soft horse. The difference is immense. Getting a soft horse however requires the rider to take responsibility too. Let the horse go left. Why not? Don't think like humans.
Firstly horses push, we pull. Our instinct is to fight and pull. Why fight, its a ridiculous waste of energy and interrupts any chance of riding well plus its just another battle between you and the horse.
If they go left keep them going left. Push them the way they offered, bring them back round to the straight, then on the correct footfall ask right again. Its like the difference between Karate and Aikido or Wing Tsun. Karate is all straight blocks and strikes. Aikido and Wing Chun meet the energy, which actually means absorbing a little of it, then redirect it to where you want to go. Horses do it to each other all the time.