Hello tkvall,
No, you have not missed anything. A couple years ago I spent many hours trying to make a menu open on command from within an application, but Apple provides no way. That Status Item up there is a menu, like any other menu, so there would be no way to
click that from within the app either.
One way that
should work is to write an AppleScript similar to
the one given in this post. You could trigger an AppleScript with a keyboard shortcut, so it would be very cool. But alas that does not work either, probably for the same reason as why hitting the F8 key (if you've enabled it in System Preferences > Keynoard > Shortcuts > Keyboard > Move focus to Status Menus) only works for Menu Extras presented by Apple apps (Time Machine, WiFi, Battery, Clock, etc.) but not those (technically called Status Items) presented by non-Apple apps like BookMacster.
It's one of those annoyances that we are forced to live with by choosing the platform that is good and reliable but stubborn.
All you can do is send an email to Tim Cook (president of Apple) and ask him to please read and heed Apple Bug 15499730 which I titled
Accessibility: Blind or Mouseless users cannot use Status Items and filed in 2013 on Nov 18. But you see the "15" prefix on the bug number is 15
million, and the last time I looked they were up to 22 million.