I think the stereotypical definition is way off base and most people misunderstand what is to be an "alpha male". I am going to give you the best definition I have come across and you tell me whether you think it is a goal you can work towards.
A true alpha’s state of mind is one of indifference, charm, humor, humility, courage and a belief that eventually, things will work out in their favor. Alphas set, keep and hold other people accountable to their boundaries. They stand up for what they believe in and don’t compromise their principles or values for ANYONE. They have an emotionally compelling vision of what kind of life and lifestyle they want to create and then resolve to pay the price, no matter how long it takes to make it a reality.”
That's a good definition that goes beyond the simple, meathead image I would typically connect with "alpha."
So are you setting and holding W accountable to your boundaries? Are you compromising your principles or values to her? These are questions to ponder.