That neck stiffness could definetly be caused by the bit...it sounds like classic full-cheek stiffness, although neck stretching exercises are good too. You can try this one bit called a fulmer...I've never used it, but it's like a combination of a loose-ring and a full-cheek. Instead of the rein attatching to a fixed metal ring, the part that you hook the cheekpiece to is a loose-ring, just with the full-cheek arms. Wild_spot is right that you don't want to switch bits too often if you don't have to, but in your case I think it would really benefit your horse to try some different things. I would definetly move to a loose-ring or fulmer or some other bit without a "fixed" end like a D-ring or eggbutt, to encourage him to salivate and relax and bend. Copper mouthpieces are supposed to help them salivate. You could also get a chiropracter out to look at him...he might be "out" somewhere in his neck and/or back that is causing him to not want to bend. You can do "carrot stretches", where you have a carrot or some other treat in your hand, and you let him smell it and then put it by his barrel, a little behind his shoulder, so he has to stretch to get the treat. He might not be able to go all the way to his barrel at first, but the idea is to get him to stretch his neck. You do it on both sides, every day before you ride, to get him to loosen up. Then you can do it while you're mounted if he starts being stiff...you give and take steadily with one rein, leaving the other pretty much slack, until he bends the way you're asking. Then you give him the rein back. You gradually ask for more and more bend until you can get his nose touching your toe. Eventually you want to be able to just give a little pull on one rein, and he should bend his head and keep it bent, even when you release all rein pressure, until you ask with the other rein for him to straighten. Remember to do this on both sides also, and always give him at least a little bit of rein as reward if he gives to you and bends!
Let us know what works!