I discovered the name. Its called the fly.
The "fly" is the small, movable piece in the tumbler that is there to block off the half cock notch. It is very necessary if the gun has double set triggers.
The way it works is, when the hammer is in the fired position (down) the nose of the sear is resting against the outside of the tumbler. It is held there by the force of the sear spring.
If you slowly cock the hammer, the nose of the sear will bump into the fly, pushing it back and exposing the half cock notch. You will hear the sear snap into the half cock notch when this happens. If you slowly release the pressure on the hammer letting it move slightly downward, the nose of the sear will fall into the half cock notch and stop the downward movement of the hammer. The gun is now in the "half cock" position.
If you then cock the hammer further towards the full cock position, the nose of the sear will pull out of the half cock notch and ride over the top of the fly allowing the hammer to move further towards the full cock notch.
When the nose of the sear reaches the full cock notch, it should snap into the notch, preventing the hammer from falling until the trigger is pulled or the set triggers are released.
If the gun is being fired with the set triggers, pulling the front trigger will release the spring loaded rear trigger. The rear trigger will then move rapidly forward momentarily driving the blade on the top of it into the sear arm. The impact of the blade on the sear will
temporarily knock the nose of the sear out of the full cock notch allowing the hammer to fall.
Almost as soon as the hammer is released, the sear spring will push the nose of the sear back against the outside of the, now rotating tumbler.
As the tumbler rotates and the hammer is falling, the half cock notch will be rapidly approaching the nose of the sear. If there is nothing there to stop it, the nose of the sear will go into the half cock notch, stopping the hammer's fall. That is when the fly does its job.
As the nose of the sear hits the fly it rotates it forward, covering the half cock notch so the sear can't enter it.
Faced with the fly being in the way, the nose of the sear rides up, over the fly, jumping over the half cock notch and allowing the tumbler to continue to rotate until the hammer hits the cap or the flint hits the frizzen.
Now, after reading this and thinking about it you can see why the fly is important if the gun has set triggers.
For guns that do not have set triggers, the fly isn't needed.
That is because, the amount of force a finger needs to cause the sear to be released from the full cock notch is great enough to overcome the pressure of the sear spring and keep the nose of the sear well away from the outside of the rotating tumbler as the hammer falls. With the finger still pulling on the trigger there is little chance of the nose of the sear being anywhere close to the half cock notch as it whizzes past.
As for your lock not going to full cock, I have seen cases where a newly installed mainspring was slightly too long and it hung up on the front side of the tumbler when the tumbler was being cocked.
If this is happening some slow, careful grinding of the material on the very end of the spring where it is hitting the tumbler can fix the problem.
Another "fix" is to remove just a little bit of material from the outside of the tumbler right above the part that sticks out of the tumbler that the spring is pushing against.
This is a bit safer than trying to grind on the spring because any scratches that are left on the spring can cause a "stress riser" that will eventually crack.
In any case, if either the spring or the tumbler are ground on you should make every effort to remove any traces of scratches that are on the ground surface.
