Nop not allowed. You have to use a curb bit. Only type of hack you can use is a Bosel and only with 3yo and only in certain classes that are snaffle/bosel legal classes. Which are harder and harder to find.
Now with all that being said. Green reiners is a Affiliate level class and not recognized by NRHA like other classes so each affiliate can make their own rules for the class. However they rarely to never deviate from what the NRHA dose when it comes to judging and equipment. Also there are certain standards that are set for that class. Bits could be one of them.
Ive heard some cruel stuff about curb bits! Some people wire there horses teeth!
That's why they now have to take of the bridle to prove it. I would never do that to my horse. I practice in a snaffle bit or hackmore, still saving up money for curb...
There is more to dropping the bit than just making sure its not some sort of cruel wiring device. They do it primarily to make sure what you are riding in is a curb, and not some other type of bit. You can have the shanks without the mouthpiece.
And in the NRHA, at least around here, you will never see any sort of cruelty with the bit.
Dropping bit to the judge has been part of the NRHA pattern for as long as there has been an NRHA pattern from what I understand. At least as long as I have been showing NRHA. It also has many other reason for it. They check for ANY mark on the horse anywhere not just the mouth. They will DQ a horse for a scratch.