Is the pressure switch still going to be in line? If so, how about closing the valve temporarily when you want to flush.
If not, how about monitoring the pressure gauge in the house to get 30 psi or more while you close, or partially close, the valve at the well.
Here is a thought experiment. Anchor a bar of soap in a 20 ft pipe. You let it sit for a few hours. You start running water out of the pipe. Water got cloudy around the soap, and after a while you see a stretch of cloud water, and then not-cloudy. There is still some soap while running the water, but it is more dilute.
Turn off the water, and wait a few hours. Repeat.
Now if you just ran the water constantly for a few days, that soap is going to dissolve even faster than if you rest between runnings. But the soap is so diluted in the continuous stream, that you no longer see soap in the stream.
Is this model applicable to your clay thing, or is it not similar and you need the cycling? I am not taking bets.