Couldn't leave a comment for some reason, so I can only write about the topic here.
As it was pointed out the mod will not work properly alongside Better Barbershop and its EK2 patch. I got around this by combining the files of this mod with the base EK2 mod
I added all the unique files added by this mod to the base EK2 folders
I replaced the base EK2 ek_falmer.txt with the modded version in common/culture/cultures
For other file conflicts, I skipped them and instead opened this mod's and the base EK2's files and manually added the falmer related content into the base mod's files
I also manually added the falmer-related contents of 00_custom_clothes.txt (found in gfx/portraits/portrait_modifiers) to the EK2-BB compatch's 01_custom_clothes.txt file (same path as above)
(After this I no longer need to activate this mod in the launcher, as all of its content is now included in the base EK2 mod and the Compatch)

It works in game and now all the falmer armour pieces show up in the barbershop and get equipped properly by other characters too, but there's a single problem that arose, that being that the women's breasts clip through the armour.
I can solve this problem in two ways:
1.: hide the clipping breasts with a cloak
2.: reduce the given woman's bust size in the BB window until they no longer clip
But it would be best if I could get rid of the bug in the first place. Could you tell me what I need to modify in the mod files to fix this? I couldn't find it on my own.