Okay, I have your answer. It took me back to my modding days, and took a little time, so hopefully you're still reading this. Also, just so we have it actually written down. THIS IS NOT pirating, and was done quite a bit prior in the days when all we had was the Demo. I went out and bought the game the day it was released, so forgive me if I didn't actually recall how it was done. Now that I've actually sniffed around the files, the solution to your question is actually quite easy.
There are only three ships available in the demo on the default settings. These are the Galaxy, the BirdofPrey, and the Warbird. In order to fly another ship go to
C:\Program Files\Activision\Star Trek Bridge Commander Demo\scripts\ships
and there you will find 2kb files with ship and bridge names. You simply change the name of the ship to the Galaxy, and you'll have that ship when you choose the Galaxy class in the demo.
You will also have to change the name of the Galaxy to whatever the ship you are replacing it with.
So for example, say I wanted to play an Akira class. I'd change (use the rename file function) the Akira.pyc to read Galaxy.pyc and the Galaxy.pyc to read Akria.pyc then go in game and select the Galaxy class.
I believe in this manner you can also play the Sovereign class bridge.
You can also change the Warbird class if you so desire, with the same method. Hence you could get two different types of Federation ships at a time even.
I would hesitate on changing the BirdofPrey.pyc as due to the name, it's the easiest to mess up. make sure all your spelling stays the same.
All this is is modding the files in the game very slightly which means be careful so you don't royally screw up your game.
I actually d/l'ed the demo for this again just so I could help you, aren't we all so nice?