Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that your puppy Tilly has a great appetite, yet a too lean physique.
I assume that she has had stool samples checked to make sure that parasites aren't part of her problem. Puppies are more likely to have parasites and if he is going out and eating things, as puppies often do, she can be reinfesting herself with parasites. Make sure these stool checks are sent out to the laboratory as they can do a more thorough check for less common parasites. Perhaps worming her with a broad spectrum wormer like fenbendazole (Panacur) would be helpful. That medication treats giardia.
Are her stools relatively loose?
If so it will be important to describe what sort of loose stools she has to try and localize the problem. Loose stools with mucous or bright red blood point more towards large bowel diarrhea or colitis, whereas just watery stools with no mucous point more toward small bowel disease. Increased frequency points toward large bowel diarrhea, whereas more normal frequency and normal or increased amounts points toward small bowel diarrhea.
Diarrhea can also lead to bacterial overgrowth in the gut. Probiotics such as Fortiflora, Proviable, or Benebac can help replace appropriate bacteria. Have you tried those?
It is quite possible that she has developed a food allergy/sensitivity and that she needs a different puppy food. Stay away from foods that have the same protein (meats, eggs) and carbohydrate (starches like rice or wheat) ingredients as her current puppy food as some of those may be the ingredients she is likely reacting to. Ideally if we suspect a food allergy/sensitivity she would be fed a true hypoallergenic prescription food, such as Hills z/d or Purina Pro Plan Veterinary Diets HA.
There are other possibilities for poor weight gain as well.
Pancreatic insufficiency is another possibility, and we see this disease more commonly in some breeds such as Shepherds. These dogs have a pancreas that produces a decreased amount of digestive enzymes, and the amount produced can wax and wane in some cases, especially early in the disease process. Testing is by running a blood test called a TLI which checks for digestive enzymes. Treatment is replacement of digestive enzymes at each meal. An easier to digest food would be expected to create less problems with digestion and as such less diarrhea.
Kidney and liver disease can cause poor weight gain too, but I would expect her to be sicker in general, eating poorly and she would be quite young for that unless she has congenital organ disease. I would expect that to have been picked up on routine blood testing and I suspect that has been done, so this seems less likely.
Dogs with thyroid disease get poor thyroid function and gain weight rather than are too thin, so that seems unlikely with her.
Primary intestinal disease, inflammatory bowel disease, or protein losing enteritis is another possibility.
In short because this has been ongoing for a bit then I would worm her, consider changes in her diet, and more diagnostics should be done.
The diagnostics can be as simple as fecal checks and cultures to start. If that isn't enough then a complete blood count and biochemistry profile (I think these have been done?), cobamalin and folate levels (to look for intestinal disease) and testing for pancreatic insufficiency to assess general health.
Given her breed if finances are tight and you need to pick which test to do first I highly recommend a blood test called a TLI looking for pancreatic insufficiency.
If these tests are not diagnostic and this truly becomes chronic then biopsies of her gastrointestinal tract to look for inflammatory bowel disease or infiltrative cancers such as lymphoma (though cancer is less likely at her age it is possible) would be the next reasonable step.
Please let me know if you have any further questions.