Always, always, always remember that a bit is just a tool. Don't be afraid to use to tool on your horse, learn how to use it correctly from someone knowledgeable, and your horse will be accepting.
You can find a medium port bit, but just make sure that it is jointed to the shanks. You don't want a stiff bar in his mouth, because this does not encourage flexion left and right, and your daughter will find herself getting more and more frustrated with him when trying to tip his nose to turn with a neckrein. This is what you should avoid: http://www.equestriancollections.com...PF00010R-a.jpg
see how it's solid from the mouthpiece to the shanks?
Look for a jointed bit like this: http://s7d5.scene7.com/is/image/Eque...R-a?$oldimage$
Your horse is plenty old enough to transition from a snaffle.
Snaffle bits are best for starting a horse, but something to remember, a snaffle carries a 1:1 ratio of pressure. One pound of pressure with your hands is one pound of pressure on the bit. If your horse is in a snaffle, the horse should hold and carry the bit in his mouth. What I mean by this is that the bit should be loose enough to have no contact in his mouth when you have no contact on the reins. It should leave no wrinkles in the corners of his mouth when the bit is in place. If there are wrinkles, there is already pressure on his mouth, and no way to measure how much. There could already be 5lbs of pressure on his mouth, so when you apply 1lb of pressure with your hands, it is moving up to 6lbs of pressure in his mouth. It is easier for your horse to distinguish the difference between no pressure and 1lb of pressure than it is for him to distinguish between 5lbs and 6lbs in his mouth, until he can distinguish 0:1 first, he most likely won't recognize 5:6 as a cue.
These ported bits, which, my husband moves a colt off a snaffle onto a high port correctional after 6 months, or 120 rides. When you put a ported bit, or any other solid bit in his mouth (that is not snaffle or broken) you want 2-3 wrinkles in the corner of the lips, so the headstall carries the bit, and not your horse's mouth. This allows the bit to stay in the same place, and not bounce around when your horse's tongue goes over and under it. It will place it correctly on the palet, and will be in the same place every time. The bit in the following picture is the one that I suggest. http://www.horsetackinternational.co...art-view-2.jpg
This bit not only encourages vertical flexion at the poll and neck, but also encourages lateral flexion left and right as well. Since your daughter will be showing western pleasure, her hand will need to be down at all times except backing. This bit will allow her to have control with a dropped and loose rein, and help your horse become more and more soft. Pretty soon, your daughter will be able to give minor cues, just a flick or twist of the wrist, and she will have your horse's ear.
Just remember, the higher the port, the softer her hands will need to be. If your trainer is working on neckreining him, it may be an ok idea to head to your tack shop and talk to the salesperson there about a medium port with jointed shanks. Your trainer can get him accustomed to the switch, as well as teach your daughter how to use the bit. She can show her how it works in the horse's mouth, and what each movement of her hands encourages him to do. Since he's as broke and quiet as you say, there should be absolutely no problem with the switch. He will toss his head and play with it for a few days, but that's because he's trying to figure out what this new toy is! My husband will leave the horse to stand tied in a halter with the new bit in his mouth for an hour or two, two or three days in a row before he rides in it. (After standing tied, he pulls that bit, inserts the snaffle and continues riding) The horse probably will fight it, but that's only because he hasn't figured it out yet. He'll get it, and it'll really do great things for him!