This should be designated as a Byway Open to All Traffic (BOAT), rather than a Bridleway. Not 100% sure, but I believe it requires changing to the following:

highway=*
foot=designated
horse=designated
bicycle=designated
designation=byway_open_to_all_traffic

See https://wiki.openstreetmap.org/wiki/Tag:designation%3Dbyway_open_to_all_traffic for further information.