Differential

I don't believe the differential is serviceable, other than replacement. The procedure is rather involved, and a lift is adviseable. Here is the literary portion of the instructions, reply if you would like the pictures, there are a lot of them.

Rear Axle - Replacement

Differential Replacement

Removal Procedure

Set the parking brake.
Raise and support the vehicle. Refer to Vehicle Lifting.
Remove the right rear tire and wheel assembly.
Place a drain pan beneath the differential.

Remove the differential drain plug (2).
Drain the rear differential gear oil.
Disconnect the electrical connector from the clutch pump check valve.
Remove the right rear wheel drive shaft.
Remove the front propeller shaft.
Place an adjustable support beneath the torque tube.

Loosen, but do not remove the torque tube-to-bracket through bolt (7) and nut (1).
Remove the bolts (5) from the torque tube bracket (4).

Remove the differential carrier-to-cradle mounting bolts (5), nuts (1), washers (2), and mounts (3, 4) from the differential.
Important: During the removal of the wheel drive shaft, the differential output shaft may become disengaged from the differential. If this occurs, firmly grasp and separate the output shaft from the wheel drive shaft. Align the splines on the output shaft to the differential and reposition the output shaft to the differential.

While simultaneously moving the differential assembly to the right side of the vehicle, disengage the left wheel drive shaft from the differential.
Remove the rear differential and torque tube as an assembly.

Remove the torque tube (1) from the differential (2).
Installation Procedure

Install the torque tube (1) to the differential.

Install the rear differential and torque tube assembly to the suspension cradle. Simultaneously guide the left wheel drive shaft onto the differential output shaft while positioning the differential assembly to the suspension cradle.
Place an adjustable support under the torque tube.
Ensure that the left wheel drive shaft is fully engaged to the differential output shaft.
Notice: Refer to Fastener Notice in Service Precautions.

Install the differential carrier mounts (3, 4), washers (2), bolts (5), and nuts (1) to the differential. Tighten the differential carrier-to-cradle mounting bolts to 50 Nm (37 ft. Lbs.).

Install the torque tube bracket-to-body bolts (5).
Tighten the torque tube bracket-to-body bolts to 55 Nm (41 ft. Lbs.).
Tighten the torque tube-to-bracket through bolt and nut to 64 Nm (47 ft. Lbs.).
Install the front propeller shaft.
Install the right rear wheel drive shaft.
Install the right rear tire and wheel assembly.

Install the differential drain plug (2) and gasket to the differential. Tighten the drain plug to 32 Nm (23 ft. Lbs.).
Remove the differential fill plug (1).
Fill the axle with synthetic gear oil.
Inspect the differential oil level to ensure it is even with, to no lower than, 6 mm (0.25 inch) below the opening of the fill hole.
Install the differential fill plug (1) and gasket. Tighten the fill plug to 32 Nm (23 ft. Lbs.).
Connect the electrical connector to the clutch pump check valve.
Remove the adjustable support from the torque tube.
Lower the vehicle.
Operate the vehicle making tight left, then right turns in order to engage the all-wheel-drive system and distribute the gear oil throughout the differential.
Raise and support the vehicle. Refer to Vehicle Lifting.
Remove the differential fill plug,
Fill the axle with synthetic gear oil.
Inspect the differential oil level to ensure it is even with, to no lower than, 6 mm (0.25 inch) below the opening of the fill hole.
Install the differential fill plug and gasket. Tighten the fill plug to 32 Nm (23 ft. Lbs.).
Lower the vehicle.

GM is full of customer-unfriendly business practices that drain their customers' wallets. Ya gotta wonder what problem they found a solution for that all the other rear axles in the last 100 years didn't do. Too many of their inventions are experiments that are dumped in their vehicles, then the owners get to do the testing for them. I had one of those experimental cars.

I'd be a little nervous about installing a used differential. Being such a common problem, why buy another one that is likely to do the same thing? What you might consider is using regular gear lube for the flush, then switch to the unnecessarily-expensive stuff.

This is the first application I've ever heard of where a flush was required to prevent problems or to insure proper operation. I have no reason to doubt BBAASPENCER, but if removing contaminants is the only reason for doing the flush, I would desire to use less-expensive gear lube for that, then use the right stuff for the final fill. There's multiple different types of locking rear axles. Those that use steel or fiber clutch plates usually require a whale oil additives to stop them from chattering when going around corners. The two tires turn at different speeds, and it's those clutch plates that are trying to keep them turning at the same speed.

Unless someone can explain that using regular gear lube for the flush is a bad idea, you might ask at the dealer's parts department for clarification. The people there will have the training on their products, and they will know what the mechanics ask for on each repair order. If their mechanics use the expensive fluid for the flush, I'd assume that is only to make more money, unless they can give me a valid reason that makes sense.

Please let us know what happens.

Cheers, Ken

Dandy. Happy to hear it's solved.