Hi
Congrats on your little Shih Tzu! She's so lucky that you rescued her! I hope she gets into good health soon! Has she had a thorough vet check up? Tzus are prone to many health issues: kidney, liver, thyroid. And given that she is a rescue dog, you may want to make sure the Vet especially checks to make sure she is fine in those areas.
I feed my dog a homecooked diet. It's all organic. I use Lew Olsen's "cancer diet" recipe, because it is the closest to a non-grain, RAW type diet for healthy dogs (not just for cancer dogs. my dog doesn't have cancer, but this diet has been great for her. All her bloodwork has come back perfect.). The diet consists of only meats, egg, cottage cheese, yogurt, and low glycemic veggies + supplements.
You can find the recipe here: [link
http://b-naturals.com/Jun2004.php[/link] >http://b-naturals.com/Jun2004.php]http://b-naturals.com/Jun2004.php[/link] FYI: there are many special diets on this site (for dogs with liver disease, kidney dis, heart dis, etc as well as many wellness articles with lots of info).
It's easy to follow. The only adjustment I make is to give the Immnune Blend at only HALF dose. And I do lessen the liver amount. Instead of 4 oz of liver per recipe, I use a BRAVO! frozen organ meat "blend", that has liver, heart and kidneys.

The liver amount in that recipe is a bit high, as it is written. I also feed it RAW (the organ meat). The rest of the meat for the recipe, I only cook to rare - med rare. You may want to start out cooking it a bit more at first, until your dog's immune system is stronger. All the needed supplements (incl Fish body oil, etc) and incl the calcium, is noted at the end of each recipe. Many dogs are allergic to beef, which is why I suggested you don't feed the beef recipe. Many dogs are also allergic to dairy, so I would skip the yogurt for now. I'd also stay away from fruits or any other sugary foods, as they can make allergies worse (and your dog's staining seems to indicate some type of allergy, yeast or other gut related issues). I think the chicken recipe and maybe a fish recipe would be good. Pork is included in the recipes, but I haven't fed pork yet. I've been using these recipes for my Tzu for over 2 yrs, and she is doing great. We boil and drain all veggies, for the most part.
In addition to the homecooked recipes, calcium and other supplements, I'd also make sure you give your dog a lot of bottled water to drink (we use FIJI brand, but any high quality spring water would be good, provided your dog doesn't have kidney issues). And the MAIN thing I'd give this dog (regardless of what food you end up feeding), is a good
PROBIOTIC, at least 1 time per day (or 2 x's per day for at least a few months). This will help replenish the good bacteria in her gut. You might also want to give some
L-Glutamine supplement with meals. This can be purchased on the b-naturals site, too. L - Glut helps to heal the gut..which is probably irritated if this dog has food allergies. The Immnue Blend supplement contains a bit of probiotic, but get a full probiotic product. You can email me if you want to know what I use. Immune Blend also contains a digestive enzyme, which will also benefit your dog.
The recipe lasts my Tzu (14 lbs) for FIVE days. She eats 3 times per day. I just divvy out 1 recipe into 15 mini portions. I have 4 oz mini tupperware tubs that I put the food in. I then store it in the meat drawer of the fridge. I sometimes make a BIG batch of 4 recipes worth. Then, I add in the calcium and I freeze each recipe portion in a ziplock. I thaw in fridge overnight, then ADD in all the supplements, and then divvy into the tubs. It's really easy to do!
After you feed the fresh food diet for a few months (I'd give it at least 6 mos), you can then maybe switch to RAW, if you wanted to. You might want to start gradually, by cooking the meats a bit less each time. There are many RAW feeder groups you can join, for more info. There are also many members here who can guide you about how to feed RAW, correctly. Or, you might want to go to kibble with a bit of fresh meats added. But for now, I think a 100% fresh organic diet is really key to getting your Tzu to wellness. JMO. There are also some herbal supplements that can help to BOOST her immune system. The easiest to use is Tasha's Olive Leaf Extract. You can just squirt 1/4 dropper into her mouth, 2 x's per day, 6 days per week...OFF 1 day. You can continue this for 6 weeks. Then, off for 1 week. Then repeat. Again, JMO. You can find the tincture at the same site where the recipe and the other supplements are.
Good luck!
hth!
Pam