Your bit selection is going to be a big indicator of your success in introducing both your horse and you to the double bridle.
Remember there is going to be a lot of metal in the horse's mouth. For this reason you want to really carefully select a bridoon snaffle that is quite thin, and a bit wider than your normal snaffle (to allow more room for the curb). I have a kk type snaffle for normal type riding, but use a french link in my double because there is less bulk to it, but use what your horse likes best (mines not very picky).
For the curb, you will likely go through 2-4 curbs before you find the one to work in, but in stages! A good starter curb is going to be quite thick, with short shanks
(I started Rowan in something like this Euro Steel Brass Hollow Low Port 5" Cheek Dressage Curb Bit
Then I would ride around, mostly on the snaffle, until you get the hang of 4 reins. Also, try different holds. The best hold I found for my small hands is to have the snaffle normally, and run the curb under my pinky. More traditionally, the curb would run between the index and ring fingers, which I find uncomfortable.
Then when the horse is more used to the curb, he will probably start fussing with the big hunk of metal in his mouth. From here I transitioned to a thinner curb, with the same short shanks, which Ro did not like. So then I bumped up to a mid port, mid shanked curb which he really seems to like
The other thing you need to realize when riding in the double, is that there is contact on the curb. It is light, but it is there. Any flapping of that curb rein will be penalized. The point of the curb chain is to be tight enough that the contact on the curb does not bring it past 45 degrees. A moderately tight curb chain is not cruel, and serves to stabilize the bits in the mouth.
As well, have the bridle adjusted correctly!! The bridoon hanger needs to be inside and behind the curb!! I see doubles adjusted incorrectly far too much or my liking.
These are both things that your coach should be able to help you with though.
Good luck!! It will take a lot of trial and error