Supposing I thought my country's papal nuncio owes me money and he thinks otherwise. So I sue him and he gets the case dismissed on the grounds of diplomatic immunity. Maybe I could go to Vatican City and sue him there. (?) But since the Catholic church has tribunals of the kind in which canon lawyers practice, might a local tribunal of that sort in my own neighborhood suffice? (Obviously it's not the sort of thing those courts usually handle. Nor do the courts in Vatican City in the present day usually try cases in which a defendant gets sentenced to imprisonment, and yet that happened and was in the headlines about six or seven years ago.)
It depends in part why you think he owes you money: is it about real estate in the US, it is about his role as executor or heir in an estate, does it relate to professional or commercial activity outside his official function? This is covered by the Vienna Convention on Diplomatic Relations Article 31. If so, you can sue him the regular way. Enforcement of a judgment can be a challenge, since he can't he held in contempt for ignoring the order, and the police can't enter the embassy to take property.
A theoretical alternative would be to take the case to the local Archdiocese, but Canon 366 precludes that "the seat of a pontifical legation is exempt from the power of governance of the local ordinary unless it is a question of celebrating marriages", so taking your case to the Archdiocese of Washington is pointless. Consequently, I think the case would have to be presented to the Roman Rota. But from what I can tell, unless this situation is a violation of church law, you will fail to state a cognizable claim.