select a menu item of a window instead of a process using Applescript
I must say that your question is rather elaborate and takes quite some reading and thinking ...    

But here is a version of your last but one applescript that just MIGHT work a little better for you:

activate application "Fiji"
tell application "System Events" to tell process "Fiji"
    click menu item "Quit" of menu of menu bar item "File" of menu bar 1
end tell


1) What you called "menu finder" actually is menu bar item
"File"

2) Menu bars with Apple (normally) don't belong to "window 1" (but to applications). You'd have to get a specific window to the front in order to process just that one.
But anyway, "Quitting" will close all of your windows ...