The "cycleway" is tagged as "path" with no width, with surface=compacted and smoothness=intermediate, whereas the route proposed by BRouter is all living_street with paving_stones.
If you place an additional waypoint on the "cycleway" BRouter changes the route, so it is not a connection problem.
Looking at street-level imagery, I would say that the surface/smoothness tagging of the foot-cycle way is too "pessimistic". Together with the missing width information that would explain BRouter's preference for the longer, but much more bicycle-friendly route along the living streets.
My experience with BRouter and the standard trekking bike profile is that iit often does ignore combined foot-cycle-ways tagged with the highway=path scheme. I have not looked any closer into this, just my personal experience.