There is a through route here that can clearly be seen on aerial photography and can also be found on DCCs Dorset Explorer selecting Public Transport and Unclassified Roads.  See https://gi.dorsetcouncil.gov.uk/dorsetexplorer#map=16.55/50.92560/-2.47104/0&layers=85/100/100//,1598/100/100//&basemap=1/100/100
It is fully legal for all traffic as an Unclassified Road, but is a 'green lane' and so should be annotated to keep it from being used by sat navs to route normal road users as you need a vehicle with off-road capability.  It's also shown on FMS - See https://www.findmystreet.co.uk/map?gclid=CjwKCAjw0dKXBhBPEiwA2bmObXMelI05_367BYPNCGE4hGPzRaiZM0KBb72aujq-uzngVI9zNFo1ZhoCoOUQAvD_BwE
The continuation S of the A3030 to Down Lane should also have the same attributes.