Welcome to the forum.
The snap you hear is the rear spring loaded "trigger" being released by the front trigger.
This is as it should be.
What is not as it should be is the rear trigger blade that is supposed to hit and release the sear arm is not doing the job.
This can be due to several different things.
Either the rear trigger blade is hitting a piece of wood before it hits the locks sear arm or it is not hitting it hard enough to disengage the sear from the lock tumbler's full cock notch.
The first thing to try is to loosen the lock screw(s) about 2-3 turns and lightly tap on the head. This will drive the lock slightly out of the mortise. Now, lightly snug up the screws and try firing the gun with the rear set trigger.
If the trigger and lock work like they should the problem is the end of the sear arm is bottoming out in the hole in the stock.
This can be fixed by either increasing the depth of the hole or by filing off a small amount of material from the end of the sear arm.
After checking out the locks tightness, the next thing I would do would be to remove the lock.
This can be done by removing the screw(s) that hold the lock in the stock. The screw(s) goes thru the stock from the side opposite the lock.
(Often, the barrel will need to be removed before the lock will come out of the mortise.
With the lock removed, look into the mortise and you should see a hole that the locks sear arm goes thru. (This is the metal arm that sticks out away from the internal lock parts).
Without setting the rear trigger, push it forward while looking down thru the sear arm hole in the wood.
You should see the blade on the rear trigger move up so it blocks off at least 1/2 of the hole.
If it doesn't move up that far there is some wood in the set trigger mortise that's in the way.
Remove the set trigger and figure out where the interference is. Whittle it away with a sharp knife.
Keep at it until the rear triggers blade will move up like it should.
If the rear triggers blade does move up and blocks off at least 1/2 of the hole, check out the lock.
The sear arm that is sticking out away from the inside of the lock must be capable of moving freely against the spring at the rear of the lock.
On some locks, the screws that hold the sear arm are over-tightened and this can cause it to bind.
While your there, oil the sear arm and the tumbler with a light (thin) gun oil. Sometimes, this is all that is needed to get things working.
If the sear arm moves freely against its spring pressure and the trigger blade is moving upward far enough to release the sear arm, the rear trigger spring is the last thing to check out.
This strong spring is there to drive the rear trigger blade upward with enough force to release the sear.
On many set triggers, the force the spring has is adjustable.
The adjustment may be a small screw that pushes against the bottom of the strong spring.
Try screwing the adjusting screw (if it has one) in to increase the spring tension.
On other set triggers the screw that holds the strong spring in place can be screwed in or out to increase the spring tension.
Usually, tightening the screw will increse the tension but on some set triggers the opposite is true.
In any case, try to increase the heavy trigger springs tension and then reassemble the gun and try it.
Somewhere in all of this you should be able to find the problem.
Oh. The front trigger on nearly all set triggers should release the "set" or cocked rear trigger with just a few ounces of force. Unfortunatly, due to the design, using the front trigger without the rear trigger being "set" almost always requires a LOT of force to release the lock.
Lubricating the lock, making sure the screws in the lock are not overly tightened and polishing the top of the front triggers blade is about all that can be done to make firing the gun without first setting the rear trigger easy.
Let us know what you find.
