Since they shut off after a few seconds, it could be a power or wiring issue rather than a simple bulb replacement. Checking for loose connections or a faulty relay might be a good first step. Some Mini owners have reported similar problems being caused by software glitches—have you tried resetting or checking for any system updates?
HCTRA Harris County Toll Road Authority

Hello all, I have a 2016 F55 mini, with roughly 95k miles on it. A warning recently popped up on the dash claiming low and high beam malfunction. The headlights themselves are for F56, they are LEDs with the halo and auto adjusting(I never had them replaced or upgraded and they worked perfectly before). The halo is on when first started. When I turn the headlights on, they work for around 3 seconds then the whole assembly shuts off. I tried popping the assembly open to see if there is an easily accessible bulb to change, but no luck. Has anyone else had this issue or know the fix? I would rather not bring it to a dealer where I will end up paying $500 plus. Thanks!!