Bad news. Gesture control does have the option to trigger a menu press, but it is under "root required" options.
Gesture control also has the option to implement Tasker Tasks, so I installed tasker, made a simple "press menu button" task, ran it - and wile running it, the action failed. Tasker first tried to run the keyevent as root, then ran a few javascript actions - but ultimately failed to "press menu".
The issue with the accessibility service not turning on, should be solvable by FIRST allowing User Settings/Power Management/Application Startup Management - the app to run in the background, but then I couldnt get jettoast.menubutton to display the overlay (might be a conflict with Gesture Control, or not).
So summery: Two more prominent apps agree, that you'd need root to input a home button press, and the only app on the play store that seemed to disagree, which I could find - doesnt work on the device.
If you'd like to play around with tasker - tasker on its own should be able to setup a task where, whenever you open your app of choice, it would wait for 5 seconds, then trigger a home button press. So your usecase should be solvable with tasker alone. But then tasker home button press fails.