Under the assumption that you don't care which node with the name:"John" property and the associated [:MEMBER_OF] relationship is removed, you can delete one of the nodes with:
MATCH (p1:PERSON {name:"John"})-[:MEMBER_OF]->(:CLUB {name:"BAB"})<-[:MEMBER_OF]-(p2:PERSON {name:"John"})
WHERE elementId(p1) < elementId(p2)
DETACH DELETE p1
The DETACH DELETE is needed because there can be no dangling relationships. Note that the create statement did precisely what you told it to do.