I think it's better.
highway=bridleway is used where there's no signs of motor traffic.
https://www.bing.com/maps?&cp=51.337633~-2.163893&lvl=15.111282&style=s&pi=0&v=2&sV=2&form=S00027

Those two tracks are designated as 'Byways' which OSM tags as designation=byway_open_to_all_traffic
From Wiltshire Council:
{ "type": "Feature", "properties": { "PARISH": "HILP", "TYPE": "1", "REF": "HILP21", "PATHNO": "21", "LENGTH": 841.0, "LYR_NAME": "Byway" },},

Please revert.