You are a mind reader, I yanked out that wire thing. It didn't help, but it's like a "belt and suspenders" thing, as somebody else here wrote.
I finally fixed it - it still poured gas even after several checks with the old float, old needle, and combinations thereof. I also checked which way the pin that holds in the float was oriented - to match it with the serrations in the carb holes.
However, the entire carb points slightly downwards, towards the cylinder. The float on the left raises from the cylinder side when the float bowl fills with gas, and the floats are not handed (left or right is the same, according to BMW).
I thought about it for a while and tilted the carb slightly, maybe 10 degrees (I rotated the top of the left carb away from the bike).
Seemed to work! I guess the tilt offset the downward slant. The other carb on the right is pretty level. Seems to run great.
I also used a soft rag to clean out the place where the needle goes into, and the float seemed to have a very, very tiny bend in the middle (maybe literally 1 degree), so I straightened that out. One of these things fixed it.