Chello!
::the sound of Lord K'jeran repeatedly smacking his ridged forehead against his monitor fills the room::
Why didn't I think of that? Elegant solution to the problem.
Let's add that to the list for everyone to comment on as a third possibility.
Make it:
3. No SF Chaplain Corps, but civilains are retained on board ships. They could still assist the Counselor, be a bartender, or serve in Sickbay (even today, many religious orders emphasize medical training).