That is what I did with my Micklem bridle (which I cut off the chin strap to make room for the curb chain.)
I put the bridoon hanger OVER the outside of the Micklem. It looks odd, but the horses are comfortable, the bridoon hanger does not dig into the horse’s poll, and my riding teacher does not mind because her lesson horses are comfortable.
Since I use the Fager titanium double bridle bits my bridle is not terribly heavy. When I did this with stainless steel bits sometimes I had trouble lifting the bridle high enough to put it on the horse (I am pretty weak physically.)
I do another oddity. When I was experimenting with mouthpieces for the bridoon I tried one of the regular Fager snaffle bits with the mouthpiece I wanted to try (the Bianca). It did not work too well. So I e-mailed Fager and they told me that they had switched to having the Weymouth curb WIDER than the bridoon (which went against everything I had read or been told for decades), because that they found with the traditional set up that the Weymouth interfered with the action of the bridoon. I tried with the wider Weymouth and both bits worked BETTER, my hand aids could be lighter, my contact got lighter, and the horses got happier with me even though they seemed pretty content with the normal set up.
Later on Fager made the bridoon version of the Bianca, the Alice.
Now my double bridle bits’ cheekpieces look normal from the outside and the horses are happy and comfortable. I also proved that even though I am over 70 years old and have ridden for over 50 years I can change my ways to make the horses more comfortable even though the new way is not like I had always used to ride.
My double bridle looks odd. The horses do not care. I don’t care. My riding teachers don’t care. Since I do not show the only things I care about is the horses’ comfort and making my hand aids ever lighter and more sophisticated. The horses and I communicate from the horse’s tongue to my little finger and we can get really, really refined for a lesson horse in a hunt seat lesson stable carrying a pudgy, elderly and handicapped rider.
I LOVE riding in my double bridles.