Very glad you posted the following pic, because it is part of the problem you are having with the Cock going down too far.
It looks to me like someone filed the "stop" or shoulder too much on this cock. In this pic, the "stop" or shoulder is shown facing vertically and to the left of the large flat area surrounding the square hole for the tumbler.
This "stop" or shoulder is one of two things (on a lock with an interior Bridle) that keeps the Cock from traveling too far forward. (On a lock without an interior Bridle, it is the only thing that keeps the cock from traveling too far forward.)
It looks like someone filed it too much and as shown in this pic, too much to the left.
The way to check to see if the "stop" or this shoulder of the cock correctly fits the lock plate is to strip the interior lock parts off to where you just have the Tumbler and Cock on the lock plate. You want to look at it from the inside of the lock plate.
Slowly move the cock forward until the jaws stand off maybe 1/8" or more from hitting the pan. If that "stop" or shoulder doesn't hit the top of the lock plate to keep it from going further, it has been filed too much. You can measure the distance by laying the lock down with the cock facing downward. If it has been filed too much, one way to fix it is to weld metal onto the Cock and correctly file it so the stop/shoulder does stop the forward movement of the Cock.
HOWEVER, on a lock with an interior bridle like this, the OTHER thing that keeps the Cock from traveling too far forward is the tail of the Tumbler hitting the rear of the Bridle. IOW, both this and the stop/shoulder on the Cock keep it from going too far forward.
So, after you check to see how much metal has to be welded back onto the stop/shoulder of the cock, then put the Bridle back on and screw it down using only the left Bridle Screw. That way you can see whether the Bridle is stopping the forward movement of the Cock (or not) by the rear of the Tumbler hitting it.
Gus