the plate change is easy, make sure to not drop it when you unbolt it. (scissor jack helps to take the weight off your arms as you line it up)

check the speed sensors, if they are cracked or split you found your issue.

If you want to play around before removing it, use an Oscope and see if you have an erroneous speed sensor signal (it has 2 sensors, one input and one output - both are hall effect - sharing same input circuits to switch on but the signal lines are separate)

as you have a later car, it maybe a bad shifter too, does the gear indication LCD next to the clock show the right position?

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2012 BMW X5 (Beef + Granite suspension model)

1995 E300D - The original humming machine (consumed by Flood 2017)
2000 E320 - The evolution (consumed by flood 2017)