Hi there,
It sounds like there may have been some post-op nerve root irritation there that could have reflexively involved the sacral plexus to create the incontinence. When I treat incontinence, I'm definitely including points over the sacral foramina, which in this dog is in the vicinity of the scar, since these points are considered the most effective for sphincter control in human acupuncture.
Si Miao San addresses Damp Heat in the lower burner, which would include inflammation over the sacrum, I believe, and it seems to fit your garden hose description of the pulse. It has been useful in some Addisonian patients, due to its adrenal sparing content of beta sitosterol.
Another formula that might help, and which also has the sitosterol in it, is Zhi Bai Di Huang Wan. This might still subdue the inflammation over the sacrum, but has the benefits of being Kidney Yin tonifying, which your response to treatment suggests is desirable. So I might be tempted to start with that one. Some authors view Addison's as being exclusively a Kidney Yin deficiency syndrome, which I don't agree with. But I agree some cases likely are.
Lastly, nerve irritation and inflammation can suggest a benefit of Minor Bupleurum and its derivatives. It addresses Shao Yang and Dai Mai pathology, which can create the pulse you describe. SMS and XCHT are fully compatible, if you care to give them both a try. I consider most cases of Addison's disease to arise from Shao Yang disharmony.
Hope those two or three ideas help. Please let us know how it goes!
Steve