Selecting a submenu item with AS (or JS)

Question

I'm trying to invoke a submenu item from a keystroke. The item does not appear in the "Keyboard Shortcuts..." dialog. I've looked everywhere and it's just not there. It's a plugin made by Mediawide that dynamically resizes objects and pages according you my preset specs.
So I decided to just write a GUI script to invoke the menu item and then assign a keystroke to the script. The script works fine when I invoke it from the finder but if I invoke the script from the ID scripts palette, it just flashes the Apple menu for a split second and that's it. That defeats the purpose of assigning a keycommand to it.
I think it may have something to do with the fact that it activates "System Events" which is an outside app.
Can anyone shed some light on this?
Below is the script and below that, a screenshot of the menu item.
activate application "Adobe InDesign CS4"
tell application "System Events"
          tell process "Adobe InDesign CS4"
                    tell menu bar 1
                              tell menu bar item "Window"
                                        pick
                                        tell menu 1
                                                  tell menu item "Dynamic Resize"
                                                            pick
                                                            tell menu 1
                                                                      pick menu item "Resize Document"
                                                            end tell
                                                  end tell
                                        end tell
                              end tell
                    end tell
          end tell
end tell

Answer 1

Thank you Dirk,
That's almost got me where I need to be. Your script to open the File Info menu works fine but unfortunately my changes broke it. (BTW, I notice That in your script you ID "File Information..." when the menu item is "File Info...".  Is there a rule somewhere regarding menu item names?)
I changed the submenu item to "Window" and the itemByName to "Resize Document" but I'm still getting an error on line 5, probably because it's a submenu item as illustrated in the screenshot above. The structure is actually Main>Window>Dynamic Resize>Resize Document. I just don't know where to put the "Dynamic Resize" part of the menu structure in there.
Here's what I have
(function(){
var m = app.menus.itemByName("$ID/Main").submenus.itemByName("$ID/&Window");
// $.writeln(m.menuItems.everyItem().name);
var mi = m.menuItems.itemByName("$ID/Resize Document");
var a = mi.associatedMenuAction;
a.invoke();
Thanks
RR