Hello,
If the engine ECU goes into limp mode the engine performance will be reduced, it will not misfire. The sensor at the bottom of the expansion tank is a coolant level sensor.
See the below video:
https://youtu.be/ddtVMO9ta2Y
How to replace the expansion tank on E39 BMW:
https://www.pelicanparts.com/BMW/techarticles/BMW-5-Series-E39/60-WATER-Coolant_Expansion_Tank_Removal/60-WATER-Coolant_Expansion_Tank_Removal.htm
How to bleed the cooling system, this is on a V* but will also apply to your vehicle:
https://youtu.be/J4p4PRCpOfA
Once the expansion tank has been replaced and the cooling system bled properly, it would be advisable to have a chemical test for a possible blown head gasket done.
https://www.2carpros.com/articles/head-gasket-blown-test
A misfire can be caused by a blown head gasket, faulty ignition coil or sparkplug.
In extreme cases a lack of compression can also be the cause of a misfire.
Please let us know how you get along.
Cheers, Boris
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Sunday, March 13th, 2022 AT 1:35 AM