Here is another image of Louis-Charles, which I disregarded from my original selection as the subject seems too plump and round-faced to possibly be the little prince. However, I have since found this portrait cited as evidence for Louis-Charles's characteristic dimpled chin and malformed ear. It too has recently come under the hammer - it was auctioned by Sadde of Dijon in November 2015.
A press cutting from 1965, attached to the back of the frame, supplies additional information. The original owner of the picture is said to have been Pierre-Charles Bonnefoy du Plan (1732-1824) garde-meuble to Marie-Antoinette and former custodian of the Petit-Trianon. It was subsequently acquired by a Captain Guvel, who was garrisoned in Paris, and then passed to the Lauvergnier family, who at one time owned the Galerie Vauban in Dijon. The artist is unknown though, inevitably, it has been attributed to David. Since Louis-Charles is shown with short hair, the portrait is usually taken to echo his appearance in the Temple during the final months of his life.
In 2008 the vice-president of the Cercle Louis XVII, Didier Duval, used the picture for the cover of his book Louis XVII: une affaire d'Etat: la piste de Dijon dévoilée. The book expounds yet another survivalist theory. This time Louis XVII is said to have been rescued by Robespierre and the Committee of Public Safety and taken to Auxonne outside Dijon, a convenient location for handover to the Austrians. After Thermidor Carnot, a native of nearby Nolay, and his brother, who was in charge of fortifications in the area, managed to spirit him away, first to Transylvania and then to Hungary. Here he spent part of his life, married and had a son, finally dying in 1851 - Didier Duval has even found a woman who claims to be his descendant. Unfortunately, it is not clear from the summaries, whether the portrait is somehow relevant to the argument or is simply used because it was found in Dijon.
Sadde-Dijon, Sale of 29th November 2015: Lot 158: Portrait, presumed to be Louis XVII,
Charcoal drawing with chalk highlights. 29cm x 20cm.
See also the entry on the Musée Louis XVII website: