I had a bit of a look on the bnf site
https://gallica.bnf.fr/accueil/en/content/accueil-enPretty lean pickings on Atania, M. Barbet and Lorient, but I did find out:
M. Jean Barbet registered the name Atania of use on his bicycles and bicycle parts on March 8, 1921. See higher up in the thread. Of course we was likely in business, and maybe even building Atania bicycles, before this. No mention of motorcycles.
In April 1923, tyre maker Hutchinson listed Atania as one of the makers of "bicycles, motocyclettes, and motor cycles" who used their tyres. I don't think you can read into this anything about whether or not Atania made motorcycles at the time.
On July 22, 1926, M. Barber registered the name Regulus to use on "bicycles, motocyclettes (lightweight motorcycles), and accessories. This doesn't necessarily mean he stopped using the name Atania, but he may have. Clearly he was either making, or wanted to make, motocyclettes.
In June 1928, M. Barbet passed his "cycles et motos" business on to M. Allano. The address was given as "1 pl. Morbihan" in Lorient, but I can't find this on a modern map. Lots of redevelopment in Lorient so it may have been built over.
And that's it! Best hope is finding adverts in the local newspaper.
Cheers
Leon