I agree with @Amberley about trying the double bridle.
I ride lesson horses right now though I used to own my own riding horses. With both sets of horses having the double bridle, with a curb and bridoon in their mouths, seemed to remove the existential angst that the horses had about figuring out what THIS rein aid means at THIS particular moment in our ride.
Using a double bridle with two bits in the horse’s mouth is not rocket science, the horses I ride seem to figure it out easily during the initial lesson of 30 minutes.
I did not have a riding teacher teach me how to use a double bridle, it was my first horse who did that. I did ask my riding teacher (who started out taking lessons from Gordon Wright) if she thought my seat and hands were good enough and she told me to go ahead and try it. Due to my work schedule I rarely had a lesson with her but she saw me riding a lot in her big pastures and she had a good idea of my riding abilities (not as good as most of her students.)
By the time I got with my present day riding teacher I had introduced 2 or 3 horses to the double bridle, with no help from a riding teacher or a horse trainer. As I said above, using a double bridle is not rocket science, a rider does not have to have the equivalent of a PhD to use one on a horse who never had a double bridle in their mouth before. My riding teacher had no experience teaching a student using a double bridle but I finally talked her into it.
With my present day riding teacher I have introduced the double bridle to 3 or 4 of her lesson horses (who also had no experience with the double bridle) without any problems at all. Most of these horses I had ridden before, essentially training the horse that, at least in my hands, that the bit is not an instrument of torture, relaxing the horse until the horse voluntarily reaches out for contact, and filling a gaping hole in the education of these up-down lesson horses.
Since you have ended up having to use two reins anyway this probably will be easy for you.
And you may find that your horse listens to your hand aids better than they do now because they are paying attention to your hands better than before.
One of the lesson horses I introduced the double bridle to had an iron jaw when I started riding him. I got him relatively good at contact after riding him for a few months to where I could control him better than his previous riders. Even so, with him looking so much better than before with just a snaffle, when I changed to a double bridle after many months of not riding him it took this 25 year horse maybe 3 thirty minute lessons to decide that the double bridle was the answer to all his fears and insecurities about his riders’ hands. The strength on contact I had to use to get him to obey me decreased A LOT and he became light in hand even heading toward the gate when the gate was open.
I never had to take a hard hold with the double bridle though the strength of contact increased with speed. My contact is usually an ounce or two, at speed my contact with the double bridle maybe increased to half a pound galloping across a humongous pasture with no fence in sight.
For the record nowadays I use Fager titanium bits for my double bridle (and my snaffles), the Fager Victoria Mullen mouth Weymouth curb and either the Fager Alicia(?) three piece bridoon with a titanium roller on the center link or my Fager Adam leather snaffle as the bridoon.
I keep on asking my riding teacher if her lesson horse shows any irritation from me using the double bridle and she tells me the horses look happy with it. I also ride with another lady on her heart horse, and she tells me that her horse also looks happy with my using the double bridle.
In spite of me having MS, bad balance, no proprioceptive sense, and I do not have any endurance at all (30 minutes is IT, I can go no longer riding a horse.) I do not ride these lesson horses more than once a week either, yet they manage to learn the double bridle just fine.
It does not matter if I am on contact or if I ride with sagging or loose reins. The horses just listen to my hand aids better when I use the double bridle AND they do not try to drag me around the ring or make my arms ache from desperately trying to hold them back.