Have you ever wished you could blow your cat's nose for it? Angel, with chronic upper respiratory issues lives here now and I have found that an infant snot sucking kit is a wonderful addition to the arsonal of assistants.
Gross name but, oh well. Rest assured, there's absolutely no chance of getting any snot anywhere near your mouth.
Use pure saline spray solution and be sure there are no added medicines or preservatives in it. While it doesn't cure the illness, it sure helps.