The most common cause for the fast drain battery is the FSU on the blower for HVAC. When battery gets too low too long the EWS is the main culprit of crank no start. Lots of the systems on e53 are controlled on the ground side not sure if injector is one of those but, I still would suspect two things: the EWS needs to be re-sync which of you have 5xx foxwell it should be able to do that or FPR; when wife's FPR o-ring failed not only would I get splurts of gas it also would read 50 psi but it took about 12 attempts of 10-15 seconds of cranking to get it to start after it sat a few days. Hard start cured completely after FPR replaced (turns out the 30¢ o-ring all it really needed but the car was old enough I felt fine replacing the filter/FPR).

You can verify spark with some starting fluid shot into the intake if you can find a vacuum line downstream of the MAF. I've read it's not good to spray into the MAF. I've run n62 motor for 20-30 seconds on a capful of lacquer thinner in the intake. Spark no fuel the engine will run on either of those. No ignition from starting fluid no spark that's the ECU on purpose not sending spark.

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)