If garden waste is placed in the large green bins, and the bins are placed with the wheels flush against the curb, there generally is room for a cyclist to squeeze by the bins without leaving the bike lane.
The following might be a workable solution.
1. The City informs all residents living on a street with a bike lane that effective some date they will be required to use garden waste bins; garden waste left on the street will not be collected.
2. Then effective that date, garden waste in bike lanes will not be picked up.
3. After a certain date, garden waste left in a bike lane will lead to the resident of the corresponding house being cited: -- at first, just a warning, after that, with a fine. In the case of an apartment house, the owner will be cited.
Perhaps others can suggest improvements to this suggestion, or another approach entirely.