I am wondering if it is possible to do the same in the caret insertion point.
I can perform a right click in the insertion/caret position rather easily.
tell application "System Events" to set frontApp to name of first process whose frontmost is true
tell application "System Events"
tell application process frontApp
set _selection to value of attribute "AXFocusedUIElement"
tell _selection to perform action "AXShowMenu"
end tell
end tell
But a left click is completely beyond me.