I would split it along Roman Lines:
- The Autonomous Archbishop of Kaerloiu (Gloucester) and Britannia Prima, (Southwest of England, Wales, and Cornwall), and Primate of the Isles
- Bishoprics of Britannia Secunda (North and Centre of England)
- Archbishop of Lindkolon (Lincoln)
- Et al
- Bishoprics of Maxima Caesariensis (London, East Anglia, Sussex)
- Archbishop of Lunden (London)
- Et al
- Bishoprics of Caledonia
- Archbishop of Dun Ad (Dunadd)
- Et al
- Bishoprics of Hibernia
- Archbishop of Liathdroim (Hill of Tara)
- Other Bishops
- Other Bishops in Britannia Prima
This is based on this
map [N.B. Flavia Caesariensis, is folded into Britannia Secunda, maybe a few border changes but the rest holds].
I think with this structure it more closely follows other Autonomous Churches, where as they aren't self governed they will have a most senior Bishop, but he won't be supreme. How Autonomy works is normally, this varies based on the agreement of Autonomy, but normally the highest Bishop is appointed, and then the subordinate bishops are chosen by the Primate. This basically means that the Insular Church is more similar to a formalised permanent Papal Nuncio, which would fit your tid bit of Canon.
Similarly, while
@The Professor made many more Primates, historically the Catholic Church has used Primate to mean someone who can Crown a monarch, and someone who can call National Synods. In this context I think the Aurlings would jealously guard the right to control Kingship and as a result Sovereignty, probably seeing themselves as "Primates" of the Isles. Similarly, National Synods would likely control liturgy and excommunication of Bishops. This being powers I doubt Kaerloiu would willingly give up.
Finally, on titles. Because of the Prestige of being the Primate, I believe that the Archbishop would likely claim to be Archbishop of not just a city but of a region, this being his canonical region, and he would possibly not call himself of
Britannia if only not to offend the other Archbishops.
Edit: Having done more research this isn't beyond the wit of man even inside the Roman Church because of
sui iuris which is a church that runs itself to a certain extent, and is synonymous with autonomous. The
sui iuris Churches are mainly the Eastern Catholics, but I could see that in this context it is possible and could easily make sense.
Edit 2: On the map, Hadrian's wall would be the border for Caledonia.