Edited: 2/26/2020 To clarify and add solution suggestions.
Unfortunately not in the way you would like.
The way Logitech does profile swapping and the way the LCD screen renders certain game elements are entirely unconnected.
The LCD screen interfaces with the currently running game (the executable (.exe) file running) to display information. This is implemented by the developer of the game.
Logitech's software does support an automatic profile switch feature. This is activated in the software by selecting the .exe file that is the program or game you want to trigger the profile switch when it runs. So in the case of Star Trek Online, you'd find its .exe file and add it to the Logitech software and then design the profile you'd like to use while it's active.
However, you cannot switch profiles automatically based on events taking place in the game, only based on a different game running.
What you can do however is set a key to toggle between different profiles, so you could tap a key a few times when you change mission types to activate the desired profile.
Notably, combining the profiles the user manually rotates between and automatic game profiles has led to a lot of undesired automatic switching when I've tried it, so I would reccomend setting two manual profiles (that don't activate automatically) and toggle between them as needed.