Hi Colleen,
You have made significant head way, as per the Xrays and the normal CBC. That is, you have pruned the branches and are now trying to understand what the root of the problem is. We know the root is not related to the Shao Yang or to Liver Qi and Blood stagnation, since the cough is slowly worsening despite treating them.
Generally the Shao Yang has two common roots: Dampness and Phlegm accumulation, and Yin deficiency. Dampness and Phlegm can obstruct the Triple Burner, interfering with the descent of Lung Qi. Yin deficiency can cause Shao Yang issues, by failing to attract, enfold and hold Yang internally. Often the pulse for these cases is either very weak or very buoyant. The pulse for a blocked Triple Burner is usually wiry (reflecting deep interior obstruction) and seems to match what you're seeing.
Thus, given the productive cough and the pulse, I'd try San Ren Tang next. If it doesn't get anywhere, you can try Lily Bulb Combination (Bai He Gu Jing Tang), but by doing them in this order, you will have ruled out Dampness before you try a Yin tonic, thus making it less likely you'd aggravate the dog with the Yin tonic.
Hopefully that makes sense. If there is any significant response to San Ren Tang, then the diet is the main problem, so I'm glad she is finally open to making diet changes
Good luck and let us know what happens!
Steve