Have you cleaned and checked the connections at the battery and starter? If you haven't yet, don't put a wrench on the starter connection unless you have battery disconnected. Check the connection at the boost lug as well. Check grounds as well. Start with the basics and then maybe time to get the testlight or voltmeter into play. I don't have any experience with the EWS so if it comes to that maybe someone else can chime in on how to rule that out.

Has a remote starter been installed?

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1988 325is (purchased new) sold
2004 X5 3.0 2005 X3 2.5
2008 X5 3.0 (new to me)