Double jointed bits may not be as mild as advertised.
I read an old book by an English born Austrian hussar (?) commander, “Seats and Saddles, Bits and Bitting” by Francis Dwyer.
This guy was PICKY about fitting the bits to the horses, who were ridden mostly with one hand because the other hand was wielding a weapon. For the bridoon of the double bridle he had a comment that double-jointed bridoons acted like a “twitch around the lower jaw,” causing pain.
Since I read this I have been super light handed with my bridoon rein when I use a double-jointed bridoon on my double bridle, I am using a Fager double jointed titanium snaffle with a roller in the middle. The ancient lesson horse I am riding has shown me that he prefers contact with the curb bit to contact with the bridoon, though he will accept contact with the bridoon (he loves playing with the center roller).
So if you want stronger contact, contact that the horse will willingly reach out for and keep his mouth quiet, try a single-jointed or Mullen mouth snaffle. With the Mullen mouth snaffle it really helps if the rider keeps the bit centered in the horse’s mouth and uses the outside rein to support an inside rein turning signal, then make sure to re-center the bit after the turns.
With a single-jointed bit, as long as the cheek pieces are not made so that the center joint of the mouthpiece of the bit is guaranteed to fold in such a way so it will hit/press against the horse’s palate, has been the standard bit for English riding for centuries. Regular rings, loose ring, eggbutt, D-ring, full-cheek or the driving half-cheek bit, people have been successfully riding with these single jointed snaffles for a long time including at the highest levels of competition.
You are LISTENING to the horse. Good for you! Borrow or buy a single jointed snaffle. If he improves then you can experiment with bit thickness but realize that a lot of horse do not read the advertising copy and do not like really thick bits in their mouths even if they are supposed to be milder.
If I ever use just a snaffle on the lesson horse I am riding I will probably start with a single jointed snaffle just because of the way he is reacting to my double-jointed bridoon, though he might miss playing with the center roller. I am sure with the wide variety of double jointed snaffles I own I could eventually find one he would be happy with but it might take a lot of experimentation. I’ll just start with a single-jointed snaffle first if I decide he would like a vacation from the double bridle.