You've gotten some good advice here, along with some things I'd probably leave alone if it was me. I agree with the NO martingale, NO checks thing. A check is a temporary fix that teaches the horse nothing about using himself unless it is in the most expert of hands. An overcheck would keep him from overflexing his neck, but unless you also coupled it with driving him more forward and teaching him to balance himself properly and then gently phased it out all it would do is fix the problem while you used it.
Have you considered that your shafts may be too wide? If I recall the pictures you posted a while ago your cart was pretty big on your horse. Even if the shafts are now far enough back, if they are too wide the reins will catch. Mine sometimes get stuck under the strap ends of my breastcollar when we're standing or if he massively drops a shoulder and overbends because they are too long and stick out under the keepers (something I am fixing before this show season). Terrets on the neckstrap are not seen in the breed ring, but they are a good idea and I bet you could rig something like that on a temporary basis. I thought that the extra rings would affect the play of my own reins and was reluctant to use them but now that I have I really haven't noticed any difference at all. It just helps remind me that I can't broaden my hands and direct the horse to "follow my hand" into a bend like I would a beginning western horse! LOL. Makes me a better driver.
: You could probably just secure some bailing twine around the middle of the strap and then have it drop down on each side to where you've tied it into loops. It doesn't really have to be stationary on the breastcollar, just at even heights and servicable to keep the reins up.
Get him a bit that does not have sharp edges and I think the suggestion to try a mullen is a good one. I'm not so strong on my bitting expertise but it seems to me that a mullen rests more on the tongue and will not hit the bars as strongly while holding regular contact. He may feel more comfortable moving forward into that gentler feel. Use the whip, use a strong supporting outside rein. To add one comment to the discussion of "inside hand," you are not going to pull him into a turn so much as you are going to give and release, bumping him gently and wooing him into giving his jaw to that side. The inside rein develops the bend, the outside rein supports the shoulders and determines the size of the circle by how much you give. The whip steps in to push that inside shoulder back up if the horse starts to fall into the circle. This is where a whip with a nice supple lash is easier to handle (with practice, LOL) then a stiff whip. I need a lot of practice with this kind of reining myself! I have trouble when I can't use my bodyweight and inside leg to keep the horse on the circle.
When you get it right with that outside rein you will know. I had this epiphany moment at a clinic last year where we were circling tightly in a hazard and my gelding just wasn't bending fast enough to make the corner. I'd been pulling him into it with the inside rein and of course he was overbending and his shoulder was popping out so he couldn't swing the heavy cart. (This was also a lack of forward energy. When we got straightened out the energy went straight from his haunches to his head and he started pulling into the traces with power, so we speeded up, smoothed out, and turned sharper.) Anyway, without thinking I firmed the outside rein like soft elastic and sat back on my tailbone, mentally deepening my seat and looking where we were going and all of a sudden it felt like I had physically picked up his outside shoulder with a leg aid and he coiled his haunches and started crossing his front legs over at the trot and swung into the tight turn like a pro. I said "Wow!" For a moment I understood the power of that outside rein and I try to get a little of that back each time I drive.
Hope some of that was helpful. Check the cart, hold better contact on a bit he will accept that from, teach him to work forward so he stops overflexing, and use rein terrets on the breastcollar as a safety precaution until you've resolved the basic issue.
Good thread!
Leia