First; With a standard pressure switch/pressure tank system, multiple tanks should never be in different locations. The tanks will fill at different rates and different times because of the friction loss in the pipe system. This unbalance in the filling of the tanks can cause excess cycling.
Second; The fact that you have to add air to a bladder tank, means the bladder is bad.
Third; Burning a pressure switch is the first sign that you are destroying your pump. The "click/click" you hear when the pump is rapid cycling is putting tremendous strain on your pump. It is also probably what busted the bladder in your tank.
Fourth; If your filter is downstream of the pressure tank, it is not what is causing your pump to cycle. If the gunk was clogging your pressure switch, the pump would just not come on when needed or would not go off when it should. The fact that it is "clicking" means your pressure switch is not clogged up with gunk.
Fifth; After you get good bladder tanks installed in close proximity, the pump clicking every minute or two is still too much cycling. Clicking every two minutes is still 720 cycles per day. Cycling destroys pumps, pressure tanks, pressure switches, control boxes, and causes water hammer that is hard on the pipe as well.
I would remove the neighbors pressure tank. If it is still good, you can install that tank at your house with the pressure switch. Then I would install a Cycle Stop Valve on the line from the well, before it tees off to the other house or irrigation. The CSV reduces the size of pressure tank needed, so the 42 gallon is more than enough for two houses. The CSV eliminates cycling, which will make your pressure switch and everything else last much longer.
Last; I would be prepared to replace the pump. After enough cycling that you needed to replace a couple of pressure switches, your pump is running on a wing and a prayer.