While it might be a designated bridleway, on the ground it is signposted and (just about) maintained as a shared cycle path, so should be tagged as such.

I've added designation tags for this route and horse tags for this route although I'm a bit hesitant about equestrian access along the whole route as the crossing over the A5112 is not suitable so they'd be a bit stuck or forced into the route.

I'm ambivalent about the tagging of the existing bridleway paths nearer the town (https://osm.org/way/24156137). In my opinion they don't cater as much for cyclists (cycle barriers, narrow & muddy... but still part of the NCN).