It is not necessary to use a breast pump on a dog with mastitis, and as their mammary tissue is formed very different than humans, it is unlikely the pump would work. The best action to take would be to gently apply a warm compress to the mammary glands three to four times per day, and to have her evaluated by a veterinarian to determine if antibiotics are warranted.
Sorry, I did not know that you had already had a thorough work-up performed. The breast pump is unlikely to hurt her, but you would need to monitor her closely for signs of irritation. In my experience with mastitis in dogs, the warm compresses and antibiotics generally have worked, so I am sorry to hear this has not helped your pet. In large animals, we also give antibiotics, warm compresses, and milk out the affected teats on the udder until the material is no longer abnormal in appearance (it should have the appearance of milk). If the breast pump has worked on your dog, this may help her recover a bit faster, but I honestly have not had to resort to this for other dog cases, so do not want to guarantee that it will help. I can tell you that I do not think it will harm her, just be gentle and watch for irritation from the pump rim.
I am sorry that your dog is having this problem, and that you have done a very good diagnostic work-up on her without a conclusive answer. I am glad to hear that bloodwork and x-rays have been normal (correct?), and it seems as though your vet has done a good job treating her with appropriate medications for infection. Unfortunately, due to her age and not being spayed, I think mammary cancer is very likely. If a dog is spayed prior to her first heat cycle, it eliminates the chance of her getting mammary cancer by 100%. If you wait until after the first cycle, the chances are only 60%, and if she is not spayed at all, she has a good chance at having this develop. Having said that, I know many intact female dogs that have reproduced and live to an old age without developing this cancer. I am not suggesting it was bad to breed her - on the contrary, if all purebred dogs were spayed we would not have any to reproduce, and we would no longer have such wonderful breeds like your Tibetan.
If that is what is happening here, it would be best to get her spayed and have a chain mastectomy performed. This is a big surgery, and will likely be expensive, but it is also the best chance of her living a long life with a good quality of life.
Hope that helps, and good luck.
A lot of dogs with mammary cancer can have some material produced by the mammary glands. I don't know that it would be true milk, but can have a milky appearance.
Cancer itself would not cause her to produce milk.