As shown in the pictures, skin isn't loaded when using CustomSkinLoader.
Maybe it's a skin source conflict? I dont see any config about this mod. Maybe add a config about skin source would be better.

The vanilla skin loading is used and resolving the name to a UUID from Mojang. If it doesn't load the skin, then CustomSkinLoader breaks the vanilla logic(the vanilla skin manager). Since CustomSkinLoader also has pretty broken mixins conflicting with FirstPerson, not that interested in trying to make that mod compatible.