Once the contact pattern has been disturbed, there's very little that can be done to correct the whine, especially if you're paying someone to experiment on it. If I had that situation with a vehicle I owned, I'd just drop the differential out and try to get the backlash back to where it started. All that costs is a little time and some gasket sealer if you're doing the labor yourself. The two big spanner nuts at the carrier bearings control backlash, so try turning the left one CCW one notch and tightening the right one the same amount. You'll need to loosen the big bolts holding the bearing cups in place before adjusting, and torque them afterwards to about 100 FT/LBS. Of course, since the first rear end I set up was done in 1964, I probably wouldn't have tinkered with the adjustments in the first place. What's happening there is that a relatively unworn part of the ring gear teeth is climbing over the pinion gear teeth, and trying to establish a new running fit. The gears aren't likely to fail, but they will be noisy. Invest in a set of earplugs and be patient- - - -in a few thousand miles the noise will subside somewhat, but it's not likely to ever go away entirely.
Fortunately, 4.57 gear units aren't scarce or expensive- - - -it's the 4.10 gearsets that are like hen's teeth! That's partially the fault of us dirt track racers, who bought up and wore out every 4.10 3/4 ton rear end we could find back in the late 1970's!
Jerry