You don't have a submersible pump? I was assuming you did. Maybe you mean the Schrader valve is at the casing, and you do have a submersible. How about a photo of your Schrader valve area.There is a shrader valve at the well pump itself.
A special Schrader valve called a snifter valve is used upstream of a topside check valve with air-over-water tanks.
You have a precharged diaphragm pressure tank. If you are getting doses of air coming in that snifter valve, it could do funny things with the KL and softener I suspect. To prevent that air coming in, you can put a valve cap make for tires on that snifter valve. The caps made for snifter valves don't seal.