You are both wrong!The attempt to link the McLemores in some way to the MacCullienmor is impossible, as that is not a surname but a title, and only one person - the Chief of the Clan Campbell - is entitled to wear it at any given time.I have also heard the claim that we are supposedly descended from the Stewarts of Appin, but that also is baseless, as it based on the translation of our name as MacLea Mor, the "Big Mac Lea".
McLemore is an American anglization of the Ayrshire name McIlmorrow, which in turn is derived from a western Perthshire / Invernesshire family called McGilmorie, from which one of its members moved south into old Ayrshire before 1525.The McGilmories in turn got into the Highlands by moving up the Great Glen from the western coast, where they had lived for centuries in the Hebrides.There the name was MacGillemuire in old Gaelic, which translates as "son of a follower of (the Virgin) Mary", and it was originally a title of the Kings of Man and the Isles.It became a surname when the mother of a bastard son of King Olaf the Black (d. 1237) named her son Gillemoire to make sure everyone knew who his daddy was, and his descendants became the Clan MacGillemuire.In 1346 the male line ended in an heiress who married Cedham Mac Muirich MacIan of the MacIans of Ardnamurchen, who became chief of the clan, as did their issue thereafter.These Mac Muirich (or son of Morris) chiefs by 1600 had adopted the Anglicized name Morrison, and today that is the name of our clan.Gilmore is a recognized sept of that clan (the Highland family of Invernesshire and western Perthshire), and an effort may be currently underway to have McLemore made one too (but I am not sure where that stands).The old Scottish Lowland version of the name, McIlmorrow, has apparently disappeared in Scotland, possibly being further Anglicized into McMurray or Murray. McLemore is thus a version of the name frozen in time.
I have had correspondence from the assistant to the head of Scottish Heraldy confirm to me that this is the source of the name, so it does not just rest on the research done by the late Dr. Rudy H. Leverett and myself alone.
As to tartans, we would be, as members of the Clan Morrison, entitled to wear its tartan. I have seen two different versions, an "Old" one and a "New" one; considering the family's history briefly outlined above, I would think the "Old" one would be better for us.