After searching the internet fpr months, I have found multiple sources for both Brest and Lorient. I assume many of those sites refer to each other, so I am still confused on where Corbet was laid down, but here are two from reputable websites:

Source 1: Authorised under the 1910 construction programme, the Courbets were a rather tardy reply to the monocalibre battleships already in service with the American, British and German navies. While of comparable displacement to their contemporaries, they had thinner armour, and were considered deficient as a result. Courbet was constructed at the Brest Naval Dockyard; laid down on 1 September 1910, she was launched on 23 September the following year and entered service on 19 November 1913.

Source 2: The four dreadnoughts of the Courbet class were completed as follows: Courbet at Arsenal de Lorient (laid down 1 September 1910, launched 23 September 1911, commissioned 19 November 1913); Jean Bart at Arsenal de Brest (laid down 15 November 1910, launched 22 September 1911, commissioned 19 November 1913)...

Source 2 is the author of one of the Osprey books on French Battleships, so I am sure he did his research. I would like to see some original sources, however. I suppose it is possible a hat maker in Brest could be contracted to produce hats for a ship being built in Lorient, but it is somehow a more romantic (if that is the right word) idea that the hats were made locally/where the ship was constructed.