Do you have Release Driver When Application is in Background enabled?
No it is off.

As I want to a keystroke to be my trigger, I assume I don’t need any routing and enable only my midi output.

On my Instrument track in Cubase I have “All midi inputs” selected.

Now if bome is opened on top, Cubase is opened behind it in the background and I hit “+”, bome’s event monitor gets a “Keystroke in” and a “midi out”. Now I hear the note C4 coming from the instrument in Cubase. As soon as I put my focus on the Cubase window though, bome does not get a “keystroke in”/“midi out” anymore and as expected: no sound coming from the instrument in Cubase.
So I assume the routing is fine, otherwhise I would not hear any sound coming from the instrument in Cubase?
When Bome is opened on top (and cubase behind it) and I hit “+”

When Cubase is opened on top (and bome behind it) and I hit “+”:

As soon as Cubase gets the focus, no keystrokes come through to bome’s midi translator 
When I am surfing with my browser or working in Outlook/Office or any other program on my computer, bome always gets its keystrokes, even if it is not on top of the desktop.
Now I tried something with my little midi controller. If I active my midi controller in bome’s midi inputs, no matter where bome is opened (behind or in front of cubase) it always gets the “midi in”. It really seems to be only an issue with the keystrokes.