Interestingly, looking at PRoW overlay data at https://map.atownsend.org.uk/maps/map/map.html#17/51.89484/-0.55593/H/P suggests that the council might think that https://www.openstreetmap.org/way/1161416279 actually is a public bridleway, although this doesn't mean that it's signed on the ground.  The overlay there is at https://osm.cycle.travel/rights_of_way by https://www.openstreetmap.org/user/Richard from http://www.rowmaps.com/ .