I just looked at the picture in your post, your dog I assume.
Our Westie is also a white dog, but much smaller. A big male Westie (West Highlands White Terrier - pic in my profile) would go about 20 pounds Bigger than that means overweight, fat - another struggle we have as our dog that goes closer to 22 pounds.
There have been many threads here on selecting dog food from suppliers as well as home made. You should be able to find them with the search facility.
I do not take my vet's recommendations on food, she sells Science Diet which is also sold in a "prescription grade" which she prescribed for our food sensitive Westie - it was sweet potato and venison. I moved to a higher rated Salmon and Sweet Potato None Grain, Limited Ingredients. Note the Salmon comes first, not the SP, that means Salmon is the main ingredients, and the specifics following claim no artificial colors/flavors/etc, and no grain is just that. Science Diet is silent on such important items.
I do not know the brand your vet recommends but I'd look at two issues 1) does the vet sell the food, and 2) how does it stand in any study done by dog owners, not people in the business of selling dog food.
As for no grain, I wounder somewhat as rice seems to be a popular ingredient of home made dog food, along with chicken and beef. In our dog's case No Grain seems to say No Rice, and we strongly suspect he is allergic to Chicken and Beef. This seems strange to my notion of a dog, but it seems to be the case. We avoid both of those protein sources. Given the vet recommendation for Venison and our use of Salmon and before that "Fish" which included unspecified amounts of Salmon it seems "wild" sources are the best for dogs such as Westies. I say "such as" in that I have read of other breeds having similar issues - but our Westie is our first experience with a dog that couldn't eat AnyThing - and did, and thrived, and enjoyed. Our Westie also rejects many food items I'd think are healthy for him, including green beans and carrots, both raw and cooked...these being other popular ingredients in home cooked dog food.