The BGT is not continuous through this area—see the official map here:

https://www.seattle.gov/parks/find/parks/burke-gilman-trail

Note that there are ongoing efforts to make it continuous here, but it currently offers no protection to cyclists or pedestrians.

https://www.seattle.gov/transportation/projects-and-programs/programs/bike-program/ballard-multimodal-corridor

Green Lake Trail is closed for cyclists, so it shouldn't be marked as a bike trail.

It was made pedestrian-only during the pandemic.

There is an ongoing effort to "complete the loop" along Aurora, Green Lake Drive, and Green Lake Way to make protected bike lanes:
https://www.seattle.gov/transportation/projects-and-programs/programs/bike-program/protected-bike-lanes/green-lake-outer-loop