AppleScript: Cannot click Menulet (Menu Bar 2) for Third Party App

I am trying to create an Applescript to automate an interaction with an application that has a third-party menulet (menu bar item, top right corner). The application in question is called Pomodoro One. It does not have built-in Applescript support, so I want to create a script that clicks on its menu bar and presses "Start Pomodoro".

After hours of research on the Internet and experimentation with Script Editor, plus the creation of a visual automation with Automator, I have come up with this script that SHOULD work:

tell application "Pomodoro One"
    activate
end tell
tell application "System Events"
    tell UI element "Pomodoro One"
        tell menu bar item 1 of menu bar 2
            click menu item 1 of menu 1
        end tell
    end tell
end tell

However, it does not work. Surprising, especially since the the automation in Automator works and this script is basically a paraphrasing of the Applescript that Automator converts its visual automation into. This is the error that I get, both from ScriptEditor as well as running the Applescript in Automator:

System Events got an error: Can’t get menu 1 of menu bar item 1 of menu bar 2 of UI element "Pomodoro One". Invalid index.

What I think is happening is that when Applescript attempts to click on menu bar 2 (which represents the menulet), the menu that is supposed to appear does not. As if the click is totally not happening. However, I am indeed able to click on menu bar 1 (which represents Pomodoro One's main menu bar). This is stumping why clicking on menu bar 2 yields nothing, especially since UI debugging apps like UI Browser indeed report that the menulet is represented by menu bar 2 when I do visual profiling of the UI.

I am hours into this and I feel very stuck. I am wondering if there is any way that I can get Applescript to play nice with the menulet of any third party applicaton.

EDIT

Based on this post (https://stackoverflow.com/questions/20595708/applescript-to-click-on-a-specific-icon-in-the-mac-menu-bar), I modified my script to be the following:

ignoring application responses
    tell application "System Events" to tell process "Pomodoro One"
        click menu bar item 1 of menu bar 2
    end tell
end ignoring
do shell script "killall System\\ Events"
delay 0.1
tell application "System Events" to tell process "Pomodoro One"
    tell menu bar item 1 of menu bar 2
        click menu item "Start Pomodoro" of menu 1
    end tell
end tell

This time, it results in the script infinitely spinning and eventually timing out with error -1712: System Events got an error: AppleEvent timed out. I can assure you that I have accessibility enabled, via System Preferences –> Security and Privacy –> Accessibility for both Pomodoro One and Script Editor.