Definitely not Bauer. He was in North America in 1876.
Also, there were three masked wrestlers in Paris in 1867 (the original L'Homme Masque, another L'Homme Masque at a rival arena, and a red mask). In terms of being the original L'Homme Masque Bauer was just one of several suspects and the only reason why he's in the conversion is because Bauer himself claimed to have been the masked wrestler. However, given how much of a carny Bauer was I don't know if I would believe anything he ever said.
A while back I wrote an article about 19th century French pro wrestling. Here's what I had to say about the original L'Homme Masque there:
"So, who were all these masked wrestlers in Paris in 1867-68? Which wrestlers were under the hoods? Unfortunately, there's no way to know for certain. There have been a number of names suggested over the years, but we'll never know for sure who it really was. Marseille Jr., Falcet, Folliet, Alfred Perrier, ​​​Professor Deligne, Andre Christol and Theobaud Bauer are some of the suspected names, but the main suspect seems to be a guy by the name of Charavay. Even contemporary 1867 French newspapers from several days after the first appearance of L'Homme Masque mention him as potentially being the person behind the mask and say that he was a photographer by trade who was friends with the directors of the arena. According to Professor Edmond Desbonnet, however, the matches at the Athletic Arena were actually under the direction of Charavay (i.e. Charavay was doing the matchmaking). Desbonnet then says that the masked wrestler was a masseur named Charvet, but I think this was likely a mistake. Sometimes Charavay's name would be mistakenly spelled as Charavet (even as early as September 1867) and I'm guessing Desbonnet made a similar mistake. But setting the contemporary 1867 press and Desbonnet's claims aside, another source also identified Charavay as the original masked man of Paris. In an April 1877 story on the famous wrestler Faouet in "Le Petit Journal" newspaper it's stated that L'Homme Masque was sometimes the photographer Charavay, and sometimes Marseille Jr. Reportedly, the source of this information was Faouet himself, who wrestled the original L'Homme Masque back in 1867. We'll never know for sure who the original masked wrestler in 1867 was, nor the other two masked men after him, but Charavay is the best suspect."
Sidenote: There's also this myth going around that L'Homme Masque in 1867 was the first masked wrestler in pro wrestling history. Not true. I've seen mentions of masked wrestlers in France as early as 1848. The original L'Homme Masque was just the most famous one.