Thats a strange one for sure. Sounds like the PO may have had the same problem and had to re-seat the axle as well. Could the axle joint(s) have failed in some way that is hard to duplicate with it out?

That's probably wishful thinking, so my guess would be a failing front diff, or I wonder if it is possible for a failing tcase or front driveshaft splines to cause things to vibrate badly enough to pop the axle out of the diff.

I'd start with the easy and move to the difficult as you rule stuff out. I seem to recall your car will drive without the front driveshaft? If so maybe pull the front driveshaft first, check the splines and if they (and the Tcase splines) look good then take it for a drive in 2 WD mode. No vibration would narrow it down while vibration remaining would certainly point to maybe the axle or most likely the diff.

I'm afraid none of it's easy, especially with your temps, but the front diff in my view would be the most challenging by far. Motor has to come up a few inches and the subframe needs to come down a few inches. It also weighs a ton so getting it back in laying on your back is fun. I'm not sure there is a rebuild kit for it either, but ebay pieces aren't too expensive. Notwithstanding the cold, it sounds like you could swap the diff out in a hellish day or two if you have a way to lift the motor and enough jacks/jackstands. PM me if it comes to this as I just had mine out of my 2002 E53 4.4.

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Original owner 2002 E53 X5 4.4i to 4.6i swap
2026 G80 M3 6 spd
2025 G06 X5 50e
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1972 Audi Fox
1986 Saab 900S
1996 BMW Z3
1998 BMW E36 M3 Sedan
2004 BMW E46 M3
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1993 Mopar 318 Jeep Grand Cherokee
2015 V6 Jeep Grand Cherokee