There's some rhyme and reason to it, just as there's rhyme and reason behind legal bits in dressage.
Because of the expectation of a hunter's manners and way of going, the preferred bit is a snaffle. Pelhams are allowed, but should be ridden with the curb rein loose, and an appropriate light, passive, huntery contact. And some horsepeople (including me) will be sceptical of a horse showing on the flat in a pelham, under the logic that if your horse needs that much bit to show in a flat class, he probably doesn't truly have hunter manners and way of going. Much more acceptable to show in the pelham over fences.
No elevator or leverage bit is allowed other than the pelham.
After that, fashion does come into play. The ginormous D-rings are a fad right now, because someone liked the look of them; however a full cheek or egg butt snaffle is still completely appropriate. You can ride in a loose ring; but it's just not as common as a D-ring or full cheek. Also, what makes a loose ring so good for dressage (the ability to move the bit in the horses mouth and encourage the horse to mouth and chew the bit) is not an advantage for a hunter, whose supposed to quietly accept passive contact without mouthing or chewing.
Now, to the OP's question - a horse that doesn't like a loose ring and tosses their head and fusses may do better in a double jointed snaffle like a french link or Dr. Bristol that conforms to the shape of their palate better. Since he's not fussy in a ported bit, I'm guessing that this is the case. The type of cheekpiece on the bit doesn't really matter at the momemt, you want to find a mouthpiece he's comfortable with and will accept contact with.
Good luck, and please post some photos of your progress.