If I was in the same position I would create profiles on wikitree for all eight Joseph Greens and seek assistance for each one to collate as much information as possible.
If you're lucky this will place someone in the right place at the right time.
Next I would build out all their descendants. The connection is not so far back that an autosomal DNA match is not out of the question. It may also be possible to go down the yDNA route for your known ancestor.
Reaching out to known distant cousins from this line would be critical, allowing you to triangulate onto DNA matches that can only be through Joseph Green.
Getting responses from Ancestry DNA matches can be hard, but you can always look them up in the electoral roll and send letters, to all 30 people with that name if necessary.
It will be hard yakka, probably cost money and success may not be guaranteed.
I solved a longstanding brick wall last year, pursued many avenues similar to my advice above (my situation was a little different) and ultimately got lucky.
