Ok I have a Big Tech Certification. I'll try to answer your questions, if I can. I have seen many limited-slip rear ends do this, as it is not uncommon. When you go around a corner one wheel out runs the other wheel as it is shorter around the corner on the inside as it is the outside. If the clutches are sticking or binding it will hold and release, hold and release, and cause it to pop as it releases. The inside wheel will sometimes hop as it sticks and releases. The special oil is supposed to let the clutches slip in corners. I have seen (on many occasions) a rear end do this when it was new, and stop doing it once the clutches got a little wear on them. I will assume this is an Eaton Limited Slip, and not an Auburn. If you will go to Eaton's website and check out the technical section, you will find what oil and/or additives they reccommend for their posi-units.
The back lash on the ring gear to pinion gear may be out of specifications. It should be from .007-.010" in order to operate without making noise. This is adjusted by adding or removing shims on either side of the carrier bearings. Once the back lash is determined to be correct, and the pinion as well as the carrier bearings have been ruled out as being defective, one may suspect the crush sleeve has been over torqued if the gears still make excessive noise.
If it is determined the set up is correct, and the bearings are not defective, then one must take a long hard look at the posi unit itself. One must know which posi unit you have, and that it is not a ratchet type that makes noise when you go around a corner? In some applications, noise is normal when going around a corner. If its a factory posi unit, and makes the noise you are referring to, then I would say what you have is an Eaton, and the clutches are binding. A rear end that has worn clutches, and a lot of carbon dust in between the clutches will make this noise. If the clutches are too tight, it will do it also. When the cover is removed one should see a lot of black carbon dust in the bottom of the housing if the clutches are worn out. Again I will stress it is vital to know which posi/limited slip rear end thats in your vehicle. GM used both types in their vehicles, and if you go by the tag that is on the housing bolts it should identify the correct lubricant that should be used. The additive they put in the rear end is probably all they did to it. This additive is a super slick lubricate that also has an anti-foaming agent to prevent excessive heat build up. Heat will make the oil unstable thus making the oil foam with millions of air bubbles in it, and render the oil not much good until it settles down and the air goes away. The additive should have made the rear end better, so I've got to believe the clutches are either worn, too tight, or not being lubricated properly. I'm not sure where you stand as far as your warranty is concerned, but I do know this is a problem they should fix, and you should demand to know why you still have the problem after they were supposed to have made the proper repairs. You have a right to an answer, and a right to have the repair work don't correctly as per the warranty. The exception would be if you have abused the rear end.
You don't have CV joints in your vehicle, and the clicking noise sounds as if you have a ratchet type posi unit, but I am not aware GM ever used this type of posi-tracion set up.
I don't care how many bad votes I get on this one, and you can do what you wish, but I have built many rear ends over the years, and I have seen this problem many times before. It is a very common problem with certain types of posi/limited slip differentials. This IS a problem caused by faulty lubrication 99% of the time.
Glad I could help you. Good Luck!!!!