Keeping the eye area clean and dry helps, bathing with warm water and a tiny amount of baby shampoo at least once a day and keeping the area dry in between times with the odd dab with a bit of kitchen paper. The staining is caused by bacteria which loveds the warm wet area under the eyes of certain breeds. She really needs to let you examine her face without biting, I know she's a baby so should be easy if you do it gently and regularly enough to get her used to it. I certainly would not add things to her food or water unless you have discussed it with a vet, you could cause kidney or stomach damage. I certainly would not contemplate antibiotics, unless the area gets infected. You could also keep the eye hair trimmed to stop the tears wicking down the face. You can also try smearing a tiny amount of vaseline on the hair once it is washed and dried, this can stop the hair staining.