I reckon the wiki page is pretty explicit: "Do not use this on normal paved roadways that are designated for buses only."
I think that's applicable here as bicycles are also permitted on this section (but not on the pavement) so it's important to have the bicycle=* access tags on there too. 
So, I'd go with highway=unclassified (it's not adopted), bus=private, and all the other relevant access tags