Jolly Stanesby is alleged to have been making a protest for the campaign group Fathers 4 Justice when it's claimed he scaled the side of the building.
The 48-year-old could be taken to court in the next few weeks if authorities decide to prosecute.
According to the Plymouth Herald, Peter Moore, spokesman for New Fathers 4 Justice, said: "If Devon and Cornwall Police press charges as expected we will be calling the Archbishop of Canterbury as a witness to explain why the Church of England fails to support fathers after family breakdown.
"The Archbishop of Canterbury, Justin Welby, is head of the Church of England so we hold him answerable to their policies.
"These policies played a major part in the reason why New Fathers 4 Justice chose to scale Exeter Cathedral."