I ran into a very similar sounding issue on my E85 Z4, the solution may be very cheap if you have access to some basic electrical tools.
Root Cause: Poor wire routing inside the headlight housing caused the bi-xenon solenoid wire to abrade against the sharp sheet metal of the projector. This eventually wore down to the copper and shorted solenoid + to the grounded projector housing.
This abrasion only occurred on one of my headlights but I believe both solenoids share a common power supply with built-in over-current protection. So when a fault is detected on one bi-xenon, they both shut off. This explains why they typically stay off for a while after a fault occurs, these types of power supplies tend to have an intentional delay before they can be energized again.
I also noticed that my high beams usually failed or 'fixed' themselves when going over bumps, this is likely because the shorted wire was momentarily disconnecting from the housing.
Solution:
1. locate signs of arcing in the headlight
a. likely only occurred on one side
b. removing the bulb cover should allow for inspection
2. remove affected headlight from car
a. much easier to complete repair on a work bench
3. cut out damaged section of solenoid wire and extend both solenoid wires by ~100mm
a. this allows for the wire to be safely be routed away from the projector housing
b. tin the wires individually before soldering them together, especially in a tight space like this
c. dont forget to protect the solder joints with heat shrink tubing
4. if possible, zip tie the solenoid wires to another cluster of wires to pull them out of the way
a. in my case I was also able to tuck them into a wire guide molded into the headlight
Suggestions: not related to original issue, but good practice
5. add adhesive cable tie mount to base of headlight to strain relief turn signal wires
6. zip tie halo/angel eye wires to plastic boss to strain relief
a. will need to cut off zip tie if bulb needs to be replaced
Other: In my headlight on of the turn signal wires had abraded on the HID igniter, luckily it was the ground wire so it did not cause any problems (shorting ground to ground, not an issue). While I was in there I repaired this damaged wire and routed it more safely too.
It is worth noting that my headlight is marked with the brand "Hella" I would suspect other Hella headlight modules suffer from the same poor wire routing. So this should apply to more BMWs than just my Z4, pretty sure BMW uses them extensively.
I have attached some pictures for clarity