HID wiring harnesses cannot just be 'spliced' unless you know what you are doing. The lamp supply is hundreds of volts and the ignitor is 1000s of volts. That's why the wiring should look a lot different to the normal low voltage wires used everywhere else. Even the connectors used on HID harnesses are high voltage and extremely so for the ignitor.
When others may say that one lamp works and another doesn't I get suspicious. The common unproven part left in your HID system is the wiring harness and connections. You have to treat some of it like an HT ignition system with a lamp on the end instead of a spark plug. I have a pair of those same lamps in my auxiliaries driven from a Hella non-oem ballast. I tried another lamp once and it worked fine.
I can only suggest your ballast to lamp wiring is suspect. Unfortunately because of the voltages involved it would be hard for me to tell you how to test it, because leakage of a few hundred kilohms caused by water or tracking inside a connector will stop the lamp from igniting. If tracking has already occurred, there will be a high voltage short causing the ballast to fault.
If you are an electrician or know one, I would suggest you disconnect the ballast, ignitor if separate and lamp. Then using a test Megger on the highest voltage range, check for leakage from each pin on the lamp holder to the other 2 pins.
If a 35 W lamp and lamp ballast for a 35 watt lamp are not working together, or work with some lamps and not others, I would put that down to losses in the HID wiring harness due to age, water, or breakdown of insulation. Those issues can reduce the strike voltage just like an ignition system. Internal lamp gas pressure during manufacture can vary to the extent that a low ignitor or maintained high voltage would cause the lamp to strike but the voltage losses will not sustain the arc.
Another issue is the wiring or fuses feeding the ballast have for some reason degraded and it can no longer supply the high HID ballast startup current causing the ballast to lock out. That degradation could be something as simple as a dirty fuse holder, or even a fuse that makes contact initially, but open circuits when there is a high current draw. If it is possible to temporarily wire the ballast direct to the battery terminals - try that. Also trace the wiring on the ballast d.c input side. What do BMW do, does that wiring go via a relay that has now got bad contacts? Or have BMW used wiring connectors which may have gone bad? On the d.c ballast input you are not looking for a voltage breakdown but the inability to supply up to 20 Amps as the lamp strikes. Have you carefully checked your battery terminals and cleaned them? The starter motor has its own terminal, separate from the loom terminal which needs to carry the high lamp start current. One of the protection features of most HID ballasts is to shut down quickly if the input d.c voltage drops below a threshold. Most current/voltage drop on a HID lamp ballast wiring occurs during lamp ignition.
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