You shouldn't need the angles, if you can set the joint's matrix.

I'll label your matricies from the base to the barrel as: A (base), B(joint), C(barrel base), D(barrel end)

Firstly, you have a target to find, relative to your joint:

target_dir = normalize(target_pos - B.pos())

Secondly, your barrel is offset from the joint:

offset = D * inv(B)

Thirdly, you need to apply that offset to get the correct position, which means rotating your turret into the same plane as the target and applying the inverse of the offset to the target position.

offset_pos = target_pos * look_at(target_dir) * inv(offset)

Lastly, you have your corrected target position, so you can look at it relative to world space, and multiply by your turret's bind pose to get B in the correct spot.

B = look_at(normalize(offset_pos - joint_pos)) * joint_bind_matrix

Where "look_at" if your library doesn't define it is basically building a matrix from a guess of the "up" direction combined with your desired "at" direction:

look_at(dir) {
  left = cross(up_guess, dir);
  up = cross(dir, left);
  return [at, up, left, zero_vec];
}

However, if you do actually need the angle, you can extract them from that final matrix's `at` vector.

Having gone through what you are doing now, around turret aiming, you have identified much of the problem.  But you probably need to simplify.  There are a number of spaces you need to consider.  When I did this, you need to keep local and world space in mind.  You calculate your target vector in the world space, but then apply your rotations in local space (hope I can articulate this well enough, im not that good at explaining it!)  

A) you obviously have a target vector, I like yourself want to calculate that from barrel of the gun.  This as you know is executing the matrix mutliplications from base to barrel, that is calculating barrels position.  

Base/Origin matrix -> First attachment.

First attachment to second attachment

Second attachment to barrel.

You have the location and you then can calculate from that the direction vector to the target point.  That's the first bit done.

Now, im going to assume that your first attach point rotates in say the X/Z plane.  Therefore you need to map our target vector on to that plane.   That allows you to set the direction of the connection point on that plane by getting the angle between the reference direction of the model and direction on the X/Z plane.  This leaves only the Y component for your 2nd connection point, but you need to base the elevation off the distance in the XZ plane.

The key points, because you will probably be a little confused here.  I base all my angles to rotate based off the reference direction of the model in relation to the vector in a specific plane, ala YAW for XZ and Pitch in XY.

I have working example this in my game (https://www.facebook.com/InsaneSoftware.com.au/).  The Gun turrets follow you on both static and moving models.  My code model is a little more complicated as I assign rotation axis on each connection point and it takes account of say moving ships with moving rotating turrets with accurate barrel ejection points.  But you get the idea, if you need some help just PM me.  I will try and elaborate offline.   My models also consider optimal rotation direction, blind spot (gun arc limitations) etc.

Edit: I assume your barrel ejection point as a direction Vector, This is key to how I solved my rotation problems.