Click check buttons via keyboard or script
In Brazil we call this a "gambiarra", meaning an unconventional, unprofessional, improvised, sometimes lazy, workaround, temporary or definitive, for any given problem:

1st step - positioning the window to guarantee it will be always in the same spot:



on run
    tell application "Final Cut Pro"
            activate
    end tell
    tell application "System Events"
        set position of window "Delete Tracks" of process "Final Cut Pro" to {1150, 550}
    end tell
end run



2 - Use a Pyton shell script to click the exact point:



on run
    set x to 1200
    set y to 673

    do shell script " 
/usr/bin/python &lt;&lt;END
import sys
import time
from Quartz.CoreGraphics import * 
def mouseEvent(type, posx, posy):
          theEvent = CGEventCreateMouseEvent(None, type, (posx,posy), kCGMouseButtonLeft)
          CGEventPost(kCGHIDEventTap, theEvent)
def mousemove(posx,posy):
          mouseEvent(kCGEventMouseMoved, posx,posy);
def mouseclick(posx,posy):
          mouseEvent(kCGEventLeftMouseDown, posx,posy);
          mouseEvent(kCGEventLeftMouseUp, posx,posy);
ourEvent = CGEventCreate(None); 
currentpos=CGEventGetLocation(ourEvent);             # Save current mouse position
mouseclick(" & x & "," & y & ");
mousemove(int(currentpos.x),int(currentpos.y));      # Restore mouse position
END"

end run



And voila, problem solved: