You could do something simple, appearance-wise. Make a large dock station for the backdrop, and put a single relevant player ship within it. When you hit repair, put the most damaged ship or the one with least CR, when you trade pick the ship with the highest cargo capacity, otherwise perhaps just put in a random ship from your fleet. For bonus points, there could be some limited art such as miniscule cargo haulers flying about or sparks on a ship undergoing repairs.
Went and mocked up what it might look like, I guess. Dark.Revenant, if you'd like me to take it down because it uses your Cathedral-class ship, let me know.
The station should probably be on the bottom, since that area is more likely to not be covered by something else. I'm using the Cathedral as a placeholder for the station docked at. The bottom-middle ship is currently an Atlas to match up with the trading screen. And your other ships can be scattered elsewhere behind the UI.