[Review requested]

Thanks for surveying and adding this. An alternative to mapping it as a highway=cycleway would be to use leisure=track + sport=bmx (or sport=cycling) and an appropriate value for mtb:scale=*. This should avoid potential problems of cycle and pedestrian routers trying to use it if mapped as a highway.

If the closed section of footway ( https://www.openstreetmap.org/way/884710002 )  across this area has reopened and follows the same course, you could remove the access=no tag.

There's some documentation on the wiki which might be helpful:
https://wiki.openstreetmap.org/wiki/Tag:leisure%3Dtrack
https://wiki.openstreetmap.org/wiki/Tag:sport%3Dcycling
https://wiki.openstreetmap.org/wiki/Tag:sport%3Dbmx
https://wiki.openstreetmap.org/wiki/Key:mtb:scale