Yes, turns out Software Update is under General (like on iPhones/iPads).
I found an example of scripting the activation of menu items.
do_menu is from http://www.macosxautomation.com/applescript/uiscripting/ with the addition of a delay statement that seemed necessary (adjust as needed); this does exactly what I wanted.
Indeed, the menu approach is better than faking keystrokes as I was doing before with the App Store part, because the ⌘R never worked but just caused an alert sound.
on do_menu(app_name, menu_name, menu_item)
try
-- bring the target application to the front
tell application app_name
activate
end tell
delay 2
tell application "System Events"
tell process app_name
tell menu bar 1
tell menu bar item menu_name
tell menu menu_name
click menu item menu_item
end tell
end tell
end tell
end tell
end tell
return true
on error error_message
return false
end try
end do_menu
do_menu("System Settings", "View", "Software Update")
do_menu("App Store", "Store", "Updates")
do_menu("App Store", "Store", "Reload Page")
