They're stored inside the Fallout - Sound.bsa
You can extra the files you need using FOMMs BSA tools then replace all of the files with silent versions.
You're looking for any sounds with this file path: sound\fx\npc\dog
There's a lot of sounds to replace so it might take a while.
The other option is to silence only your companion and not every other dog, you can do this by extrating all the dog anim files (.kf) from the Fallout - Meshes.bsa and duplicating the folder, renaming it, changing the sounds keys in every single .kf using nifskope and setting your dog to use the skeleton that's inside that folder instead of normal one.