My horse is like that too, very particular about the cheek pieces. I've had him in a kimblewick (off the slots, so it was basically a dee ring snaffle with a little more stability) and he HATED it. Tried loose ring, full cheek, all sorts of different mouthpieces from rubber through mullen to single jointed, double jointed with a lozenge, and he is now in an eggbutt french link. And he LOVES it. Better even than the eggbutt with a lozenge that came with him.
And yet, I had a horse that couldn't stand eggbutt... what worked best for HIM was loose ring double jointed with a lozenge, and we tried ALL the bits we tried with Monty on that particular horse, minus the french link because it was sold to me as being "the next step" from the lozenge.
We had a pony that we had in a single jointed loose ring snaffle and that worked pretty well, but after I had the success with my old Standie in a double jointed snaffle, I bought one for the pony and tried him in it... he went SO much better, responded to the bit more promptly.
I find that the double jointed mouthpiece works well for more different horses than the other kinds of mouthpiece, but as for cheekpieces, it's actually amazing. I know someone who has brilliant hands and trains her horses very well, but her arab, she can't jump in anything other than a happy mouth mullen pelham. On the flat he's in a very soft bit, can't remember what. Monty is somewhat similar, very responsive and light on the flat but gets strong while jumping - in both cases it's sheer excitement. Although in my case I'm sticking with the snaffle, simply because Monty can't cope with anything stronger than a snaffle.