You are asking a very interesting and valid question! For a more detailed description of bits and bitting, see my article on 'Bits and Bitting" on the site.
You are correct - a double jointed snaffle type bit has a less sever action on the roof of the mouth and bars to a lesser extent. I consider using this type of bit on horses with very small or fine mouths or a horse that is showing discomfort in the roof of his mouth - often they will 'yaw' at the bit. I like starting youngsters on a double joined snaffle too.
Every horse is different, and you will need to take the horse's oral conformation into account as well. If your horse is working well in the snaffle he is in, there is no real reason to change him. If he accepts his current bit and is working well, that is a sign that he is happy with his bit and the way you are riding him. Sometimes changing the horse's bit can cause more problems than fixing things, although this change is not too big. If you want to try him in the double jointed snaffle, speak to your instructor again and let him/her know that you are going to try the new bit and discuss the session once you have ended to decide what to do. Sometimes we do use two bits on a horse that tends to get 'stale' in his mouth - swopping between them every few weeks to keep him sharper off the bit.
You could also consider borrowing a bit to try before committing to buying one.
Aaah - well done - you have practically answered your own question! While the number one cause of tilting the head like you describe is teeth or a bit that is not comfortable in the mouth, your one-sided riding could very well be causing the problem. When the horse travels straight, the back feet follow the prints of the front feet. Thus even when riding a circle, the feet should be tracking over each other. When the rider has a stronger side, what often happens is that the horse becomes crooked because the rider's body is pushing the quarter skew or causing the neck to be crooked. Often the rider will be sitting with more weight down one side of the body, worsening the problem. On a circle, this is also often caused by the rider trying to bend the horse onto the circle or turn with the inside rein (we like to call it 'inside-reinitis' !) which is not the correct way to ask for bend. What that does is cause the horse to fall onto the inside shoulder and the head tilts in an effort to regain balance.
What I would suggest is that you talk to your instructor about this problem off the horse's back and ask him/her to offer you a few lessons on the lunge in order to work on your own position and balance. Once your body alignment is better, the tilting should improve. If it does not, you should be looking at the horse's balance and straightness and again, your instructor should be able to give you a range of exercises to improve that. Your horse is young, which is a both a pro and con - a pro because you can can correct any problems now before they become ingrained, and a con beacuse if you don't, it is going to affect the way your horse works for a loong time to come! (Bit of a paradox I guess!)
Good luck and do let us know if there is anything else you need help with!
The change that you are going to do is very small, so in his case I would not do anything except make sure the bit fits properly and ride him as usual - he will tell you quickly enough if he is not happy!
What you can try if ever you are making a bigger change, or want to introduce a young horse to the bit and keep things as pleasant as possible is to do what I do. When I introduce a bit, I coat it in molasses syrup and roll it in mollasses meal. Very messy - be warned! If you only have the syrup that is fine too. I then put on the bridle and bit into the mouth of the horse in the stable and leave him there to chomp away. They love it. I like to think that in this way they associate the bit with something really good before all the riding even begins. On a youngster that has not been backed yet, I repeat this every day for over a week and let them stand for about 5 to 10 minutes or until they are done licking up all the syrup. They cant wait to get the bit into their mouths!
If you wanted to coat the new bit with molasses syrup, go ahead - your horse will love it! Otherwise, don't worry too much about the change - if your hands are kind and your contact steady, your horse will hardly notice the difference!