This is just IMO because I've never heard of a dog having an iodine deficiency until I read your post and looked it up.
Do you have the name of what the vet wants you to use?
Does the supplement that you are using contain salt/sodium? No, if I read your post right? It looks like she needs .7mg of iodine per pound of food if my source is accurate.
I don't think your vet is just wanting to use her nutritionist. I think she may be honestly concerned because to just tell the owner to start adding salt could leave room for error. So I would either do what the vet wants or if you really trust your nutritionist, talk to her about the exact amount of salt that should be added.
Out of curiosity though, how did they check for this? Was it is specialized blood panel? How is her thyroid?