As usual I seem to figure these things out on my own through dumb luck.
For anyone as confused as me about this, what you need to do after logging in under Admin is to edit your profile and enter your desired display name as your first name. My problem was that I had left this blank when installing bbPress. If you save that and go back to the Edit panel you will now find that this name is included in the dropdown menu of available ‘display’ names.
Honestly, I really love bbPress so far but it sure does make things unnecessarily complicated.