I can't figure out why grass or hay up the nose wouldn't disintegrate over time with all those white cells working away at it and eventually be expelled from the nose. We don't want him under a general anesthetic, so there aren't any options for a better diagnosis, plus the nearest clinic with a scope is over 2 hours drive away.
Sounds to me like the dog requires a scope in the nose to see what's wrong. But if the dog has an infection, the antibiotics may temporarily help.
Although you think it may be grass or hay, it could be some other foreign body. OR it could indeed be a small piece of hay, that is still there. As far as I know The nose is not like the stomach which would likely have digested/disintegrated the hay.
In terms of this dog's quality of life...to my ear this dog is happy and so ending his life at this time would never be an option for me. A snotty nostril is nothing when a dog is happy.
Hope that may help and good luck!