For one, STO is its own place in the timeline, and the biggest war the Federation has ever faced, so I don't see any good arguments as to why they wouldn't use any ship they can get their hands on, and no reason not to implement it that way. I mean, if this was a game about the dominion war, sure, you could argue that the Federation didn't use any andorian ships in that, and you'd be rigtht, but it's a whole new aera in the universe so there is no reason not to mix things up a little.
Secondly, including a civilian faction would only make the content problems worse, not to mention, increase the work needed to get people flying these kinds of ships by a lot. Implementing them as ship costumes would primarely require some modeling and texturing, on the other hand, introducing a whole new faction would require huge ammounts of work, and the current factions don't even have the same ammount of content.
Thirdly, you can already play your character as a civilian if you want to. There is no rigid chain of command in the game, so if you want to just not plop a badge on your uniform, color it purple and say "I'm a mercenary" then who's going to stop you? Why not give you the option to fly an appropriate ship?
Afterall, it's not like the game forces you to use standard issue weapons either, you can lug around a Klingon blaster rifle, or some of the generic phasers the Ferengi sell you. You can even put disruptors on your federation ship, which never happened in any of the shows.