I've discovered ReactNative provides a way to inspect
DOM of application (with android, shake device, toggle inspect).
It turns out my menu item was shadowed by Context.Consumer.
When I removed <Provider> tags from my render () section,
it finally worked (was able to handle clicks).
Probably worth mentioning: from the very beginning my
AppContainer at the top most level was wrapped like this:
<PaperProvider>
<StatusBar
backgroundColor={Colors.TOOLBAR_BACKGROUND}
barStyle="light-content"
/>
<AppContainer />
</PaperProvider>