It was not uncommon or unusual for medieval priests to participate in war. /u/idjet or /u/sunagainstgold would be able to tell you more about the overall nature and structure of the medieval church, but in general, men who ranked as archbishops and other high positions within the church were not merely spiritual leaders. They had responsibilities and duties as secular lords, which is unsurprising since they had control of vast estates and incomes from their holdings. Archbishops and the like were often the second sons of aristocratic families, given positions in the church as a means of securing influence and power for their families.

David Green says in The Hundred Years War: A People's History: "Clergymen were not only victims of war; they shaped and directed the conflict...They were victims and instigators of violence, propagandists and peacemakers." Sometimes this role as "instigators" extended to direct participation in battle. The English monarchy required priests to be involved in local defense, either by active service in battle or by contributing funds to raise troops to defend coastal towns. An archbishop who fought in battle would have come arrayed as a knight in full armor, just like his secular peers of similar social rank. Even assuming that English soldiers would have been able to distinguish an archbishop in the heat of battle, they would not have had an issue in attacking or killing him in exactly the same manner as they would any other French aristocrat.