Trying harder to locate a vet is my first suggestion. Other than that, random thoughts include---
If she doesn't have an elevated temp, it's likely not bacterial, in my experience. Or if it is, it would be in the early stages. Any time I've had a cow sick enough to have a dark nasal discharge, she had an obvious fever. What makes you think it could be bacterial?
Is your cow ruminating and chewing her cud normally? Any changes in her feed recently? Some of the symptoms you describe can suggest a cow that is having digestive trouble, although the normal appetite you described really doesn't indicate that.
I know you said her manure is normal, but is there a normal volume? Is she bloated at all? Cows with digestive troubles will often cough after eating and rumen contents will come through their nose during vomiting or regurgitation, giving the appearance of dark snot. You also said that she coughs after running. Why is she running? Lastly, refer back to first suggestion.