Any good non-prescription treatment for chronic nasal inflammation?

Yes. Antihistamine/decongestant combinations may help. Nasalchrom nasal spray used 4 times a day may also help if the inflammation is caused by allergy. Do not use instant decongestant sprays you can develop rebound congestion and become dependent.
Saline rinses. Buffered saline nasal rinses have been shown to be as good, if not almost as good, as medicated intranasal steroids in relieving the subjective degree of nasal obstruction (snot-20 questionnaire). A simple mixture of sodium bicarbonate, sea salt and distilled water is all that is needed to relieve much of the nasal congestion that is felt on a daily basis. Failing this, an ENT evaluation is needed.