I was roaming around Byzantine literature on early Islam and found Constantine VII Porphyrogenitus's De administrando imperio. I also found a commentary on its sources by John Bury. Bury was a brilliant scholar, hampered only by his lack of access to Syriac material.
Now, in these days Alim and Mauias were roused up to war against one another, disputing over the rule, which of them should be lord of all Syria. They encountered one another by the river Euphrates, and joined in fierce battle one against the other. But when the battle was at its height and many were falling on either side, the multitudes of the Agarenes (Arabs) of both parties cried out: “Why is this, that we slay and are slain, and our tribe perishes from among living men? But let two elders be chosen apart from both the parties, and whomsoever they prefer, let him have the rule,” Alim and Mauias were pleased at this saying of theirs, and, drawing off from their hands their rings, which are a token of rule of the Agarenes (Arabs), they gave them to the two elders, and placed their authority at the disposal of the two elders, confirming the matter by an oath and settling it so that whomsoever the elders might prefer, he should be lord and chief of all the Saracens. The two elders entered into the middle of the battle array of the two parties, and took their stand face to face in the space between the armies; the elder of Alim was a man devout according to the nation of the Saracens, one such as they call ‘cadi’ (judge), that is, faithful and sanctified; but the elder of Mauias was devout only in appearance, but in all else deceitful and arrogant and surpassing all men in mischief. The elder of Mauias said to the elder of Alim: “Do you speak first what you will, for you are prudent and devout, and far surpassing my years.” And the elder of Alim answered thus: “I cast Alim off from the rule, as I drew his ring from his hand and drew it on to my own finger; now will I cast off the ring of Alim from my finger and therewith cast him off from his rule also.” The elder of Mauias made answer again: “I drew Mauias into the rule, as I drew his ring on to my finger; now will I draw the ring of Mauias on to his finger.” And then they parted one from the other. So Mauias took all the dominion of Syria, since all the emirs had sworn to each other, saying: “Whatever the elders say, we will be obedient to their words.” And so Alim took his army and departed to the region of Ethribos with all his kin, and there ended his life. After the death of Alim, his sons, regarding their father’s counsel as nonsense, rebelled against Mauias, and joined fierce battle with Mauias, and being worsted fled from before his face, and Mauias sent after and put them all to death. And thereafter the rule over all the Arabs came into the hands of Mauias.
The ring story was fêted all through the 1700s and 1800s, likely due to it being picked up by D'Herbelot. But Bury noted of Constantine's telling, source unknown. Bury believed there was a source. Bury had just finished tracking down Constantine's attempts to juggle other sources. If Constantine had been obliged to dig up Syriac or Arabic texts, Constantine would have told us; and Bury knew all that. So - the lost source was Greek. Can we offer Bury any help here?
I'd also seen references to Leone Caetani, Annali dell'Islam, v. 10: Dall’anno 36. al 37. H. (Milan: 1926), 59. I could not locate Annali beyond v. 8 - but I did find an English translation in a footnote by Jenkins, to De administrando imperio, 2.77: the episode of the ring is not related in its entirety by any of the Arabic sources that have reached us (instead, Masʿūdī has the turban, and the ʿIqd the sword), but the narratives of Yaʿqūbī [Ibn Wāḍiḥ] and [Abū'l-Faraj] Bar-Hebraeus presuppose it.
With Bar-Hebraeus we are back to D'Herbelot's sources. I checked Budge's Chronography; the relevant material simply apes Michael the Syrian. So it is to Mukhtasar al-Duwal we must go, Pococke's Historia Compendiosa Dynastiarum - and here it is in page 121, sicut annulum hunc manu meâ exuo.
Another very long version of the Ring story is in the 1234 Chronicle, in Syriac.