From the start, let me say I am not a fan of draw reins. For every skilled professional that uses them constructively for part of their work on the flat, I see 10 or 20 others simply pulling the horse’s head down or in towards its chest and creating a hollow, stiff, heavy, or tense horse that either pulls like a train or else puts its head even higher when the draw reins are removed. Using them while jumping I believe is even worse because, at best, they prevent the horse from using an important part of his body to keep his balance and allow him to jump easily and naturally. At worst, a horse stands the chance of putting a front foot through the draw rein–with either a serious fall or nasty yank on the bars of the mouth as punishment for the horse trying to execute a good jump with his knees up!
As to the head shaking, I’d love to see a video of exactly what your horse is doing. There are some horses afflicted with a Syndrome that causes horses to quickly snap their nose back (almost appearing to have been hit on the nose by a passing insect), repeated randomly or almost constantly. Some horses with this condition compete successfully when fitted with a loose mesh covering over their muzzle. Tension exacerbates the nose flipping, so draw reins or any other attempt to restrict your horse’s ability to move their head is not likely to help the situation. Most of these horses will do it when ridden quietly and even when turned out or on the lunge. If this description is accurate for what your horse is doing, do some research to learn more about it. Since your question indicates that your horse only does this on the last part of the approach to a fence, I find it unlikely that this is the case with your horse.
Head shaking could describe a number of different actions by a horse. Is your horse is raising his head erratically and rather stiffly or violently? Does he speed up and lose his rhythm at the same time? Or is he shaking his mouth from side to side? Are you trying to either hold him back from rushing, or attempting to ‘help’ him when you feel you’re getting a bit too deep to a jump? Rushing, with or without shaking the head, is almost always an attempt to evade excessive restriction or a rigid hold on the mouth. It doesn’t take long to make a horse defensive in those last important strides in front of a jump when a rider gives ‘instructions’ to the horse so late that it interrupts what the horse is already prepared to do. It takes far longer to convince a horse that the cause is no longer present than it did to create the resistance in the first place. You are trying to find a bit that your horse finds comfortable, but most any bit with draw reins pulling down on the bars of his mouth could make a sensitive horse, or one that has learned to anticipate the worst, try to get away. These horses are not trying to ‘get out of jumping’ but are only trying to find the freedom they feel would make the jumping easier.
Regarding tack changes, my first thought for a fussy horse would usually be a D-ring with a mouthpiece that the horse finds the most comfortable. A D-ring or egg-butt snaffle tends to sit more quietly in the horse’s mouth than a loose ring. If you use a flash noseband, be sure it is not so tight as to prevent normal chewing. I would question the ‘why’ of using a two or three ring bit with draw reins; normally the bits with multiple rings are meant to raise the horse’s head, while the draw reins are preventing them from doing so. My honest opinion is that as much as your trainer is seeking a “quick fix,” a good result on a long-term basis will mean going back to basics. You will need to start a relaxed horse that is capable of maintaining a quiet rhythm in self-carriage both in the flat and over rails and small jumps while your horse learns that a soft, non-interfering contact is something to seek and not evade. Until you have this, let your horse be responsible for adjustments at the jump for the shorter or longer take-off situations. When he learns this is part of his job, my bet is he will focus on the jump and forget the head shaking.
In the meantime, as distracting as it is to the rider when a horse’s head is moving around on the approach to a jump, it is amazing how easily a horse can jump even a very large and difficult jump even when an insect flies into an ear and produces quite violent head shaking as a result. I can remember several times on course in big classes when my horse successfully completed big tracks despite repeated head shaking as they try to rid themselves of a pesky bug!