If you were to (or had to) choose just one expansion, I'd recommend Exodus. Not because it's the better one but because it gives you more variety of what to include and what to leave out. Pegasus doesn't do that for you. If you include Ellen Tigh and the Cylon Leaders, you also have to include Treachery cards. If you include Treachery cards and the aforementioned characters, you pretty much need the Pegasus board. You can leave out the New Caprica board (which I do most of the time) but if you want to include some of Pegasus you will have to include the majority.
That said, I have to mention that the characters of Pegasus are a fun addition. I own both expansions and I wouldn't have it any other way.
About this talk of specific expansions and the core game being too one-sided:
The one thing I keep seeing at cons in both Europe and the US, in different gaming groups wherever I go, is that it really depends on who is playing. Some players are more competitive than others. Some groups are more competitive than others. Some players or groups grasp the depth of BSG (with or without expansions) much better than others. Some players will discuss possible actions more than others, some will even make a decision simply based on the fact that anyone could be a Cylon and therefore they don't trust anyone, period. So if you play with the same kind of people most of the time then, yes, your experience will likely be the same.
In my experience it strongly depends on how well you know the game and expansion(s) you're playing. If the experience is bunched up on the human side, yes, then the humans will most likely win...and vice-versa. Some players have a great poker face and some don't. There is this one player I can read like an open book. And I tend to play him like a fiddle. With other players that doesn't work so well.
What I'm trying to say is this: if you think BSG, with or without specific expansion(s), is too one-sided then go and find a different group at a local game store and play. I garantee you will have a very different experience...if this new group plays a different style.