When you change a dog food suddenly, instead of slowly crossing the dog over then your dog can experience illness such as you describe.
To me a premium dog food has little to no filler and good quality meats in it. Not by-products.
I have one dog with environmental as well as food allergies so until now we have been feeding her salmon and sweet potato but since there's been a massive recall we have switched over to a Canadian brand (made in Canada) called Nature's Harvest grain free duck and potato. It's available only at Global Pet Foods which does not sell low end pet foods. Their house brand is of excellent quality but about 30% less than name brands. What that also means is both my dogs eat it because I'm not buying two different foods. It contains duck meat as a first ingredient and contains no soy, corn, wheat. We also feed dry food only.
Your dog can be picky if you allow it. YOU choose a quality food consisting of good ingredients for a small breed dog. You're the one who knows what's best for the dog. No healthy dog will starve itself for too long when food is made available on a regular basis.
We also do not offer any crappy treats and never anything that will interfere with their meals. We offer them only freeze dried liver pieces or dehydrated sweet potatoes. I make the sweet potato treats myself in the oven. The freeze dried liver pieces that I get are Benny Bully's (or our Canadian Nature's Harvest). They are straight meat, no additives and no grains. They're low in fat and calories and I haven't met a dog or cat yet that doesn't like them. Plus they're brittle so you can snap them into smaller pieces for cats and small dogs.
As for the sweet potatoes, I slice them lengthwise 1/4" thick. I lay them on a baking sheet on parchment paper. I bake them for an hour or so at 350F then I turn the oven down to 250. After an hour I flip the sweet potatoes over and bake them for another hour. They should be moist and chewy, not crispy. My dogs go absolutely mad for them. They're natural, healthy, and high in fibre.
Also, everything I feed my dogs is approved by my vet.
If you're in the US, you can shop at Pet Smart or Petco. I would recommend Wellness, Orijen, Royal Canin, Innova, Fromm, and Natural Balance. Stay away from anything made by Diamond including Taste of the Wild and Nature's Domain.