How to run chlorine feeder at the same time well pump is running
Use a relay with a 240 coil (connected to the well pump circuit) and 120V contacts (connected to the feeder.)

Use the pressure switch only to switch relays (at any convenient coil voltage) and use (suitably rated for the power) relays to switch both devices. But the first option will be cheaper - only one relay, and that for the low-power feeder.