The wrong gear ratio on an AWD vehicle will cause bucking. I will not admit to how I know.
Most likely they have the G80 locker, and have not used Diff oil with the special friction modifier you are supposed to use with it. I do not know how much damage, or over how short a period of time damage will occur, but that may be your issue.
Other issues that this could be:
A bad U-Joint somewhere
A bad axle shaft
A bad transfer case, but I'm thinking the noise wouldn't be coming from the rear.
Same for a bad front differential
The only other thing I can think it would be would be suspension components
But I can only toss guesses against the wall and you see which one sticks, so to speak.
If you are feeling industrious, make some recordings. At 5k, a part out would bring you profit, plus whatever you get for scrapping what you don't sell.
The only other thing, in terms of a daily, is that it does have 240,000 miles on it. A maintained Suburban/Yukon Xl is going to go 300 to 500 thousand miles. You are awful near that lower limit. AND it sounds like the differential is eating itself, and it couldn't be fixed by 3 differential swaps. If I didn't have three in various stages of repair already, I might consider something like that.