I think that there are many ways to be happy. Is it possible for a polyamorous person to be happy in a monogamous relationship? Sure, anything is possible. A bisexual person can choose to commit to a woman or a man. Are they then denying a part of themselves? Not necessarily. It just depends on what makes them happy.
I, personally, see polyamory as part of who I am. I would not be happy being with a partner who wanted me to give up that side of myself. I am certainly capable of committing to just one person, I just don't want to for a variety of reasons.