Judging from the confusion and guessing in the comments, it’d probably be best if you posted a link to the area you’re asking about.
ETA based on your description of the road in question, I’d suggest cycleway=no
to indicate that there is no explicit cycling infrastructure on that stretch of road, bicycle=yes
to indicate that bicycling is allowed on that stretch of road, and tag that stretch of road as a member of the bicycle route relation.
I’m not sure about
bicycle=designated
in this case. From the wiki page that you linked: