You could probably solder in a jumper wire to by-pass the broken pin but sounds like that's out of your wheelhouse (and from what I've seen on a video might be likely that you can cause more damage to the fine wires if you open up the module) so maybe take it to a mechanic if you have a good one nearby or maybe send the module in fior rebuild??(I have no experience with that so maybe someone else can chime in on doing that.) You can't assume that the module is causing the trifecta but the broken pin needs to be delt with either way.

You might check what the broken pin is for as well, small chance it might be something redundant or not critical but it would be interesting to know.

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