So using the anti derailing yellow and green lines for the LEDs, results in the appropriate one lighting the appropriate LED. But there is a bit of residual voltage on the other LED, dimly lighting it.
As GRJ suggested above, using a relay for the LEDs is a solution. Driven by either the yellow or green line. And you also could momentarily use those lines to trigger the change in the switch as shown by Bruce...or as shown by Dennis, pulse the white wire on the DZ-2500.
That’s a 17+ dollar fix. Maybe cheaper sensitive to trigger relays can be found on ebay.