I have added comments inside some @avantronica posts to indicate stuff that is already addressed in development. One topic discussing multiple issues is otherwise hard to keep track.
Press tab to cycle between buttons, space to hit the focused button. Adding specific accelerator keys to that dialog does not appear to be low-hanging fruit.
I'm on Mavericks and I tried, no device attached at this moment, here's what I found ..
As follows ...
Launch program
Access a Helpfile from Help Menu
Close it (either via key command or mouse on window close element)
(Try to) Access Help Menu again
Close and retry
It gets confusing now, first two* attempts worked ( * may have been just once ) (exact same steps as above, no other interactions)
Embarrassing !!
The following eight attempts didn't work properly (exact same steps)
Relief !! Well, you know what I mean - issue as described originally
That's about as simple as it could be to replicate, but it's clearly a temperamental issue !
Inconsistent outcomes here is not that helpful, one to monitor perhaps !
I'll screen capture it if I remember fwiw
yeah, its temperamental I can sometimes get it to happen...
in fact on one occasion, I had ALL the menus disappear 
(i.e. no File menu or anything !)
(its seems to be something to do with the console being the only window left open... id guess some 'odd timing' between windows events/focus changing)
OSX 10.11.6