The Auring title would just be 2: Caesar of the West & Imperator of Brittania/Britannia. Bretwalda is effectively a translation of the latter, it's a reduction of Breten Anwealda meaning Broad/Britain Sole-Ruler and became a common translation for Rex Britanniae.
The Coelings would probably just recognise the Imerador as High King, sometimes as just HK of the south, sometimes as HK of all the Britons (Aurlings and Coelings alike), depending on relative strength and recent wars.