Baking powder, especially if too great a quantity is used, adds an unpleasant flavor to a baked good. Even in an appropriate quantity it can be noticeable and it certainly doesn't do anything to enhance the flavor. Many baked goods traditionally don't use a chemical leavener at all, but instead rely on technique. Creaming butter and sugar together or whipping egg whites was historically used to make cakes which rose solely based on the bubble network that was created.
Yeast, on the other hand, creates a delightful flavor that you associate with your favorite crusty loaf of bread. Yeast can be used not only in making bread but also in some excellent cakes (St. Louis Gooey Butter Cake, for example, although many "knock off" recipes cheat here and miss out on the true goodness). Yeast also provides a significantly different texture during the rising due to the intentional creation of a gluten network (usually something you absolutely don't want in a quick bread or quick cake) — you don't get a crumb with big, airy holes from baking powder or baking soda with an acid.
Also, in my experience baking powder lasts six months in the pantry and yeast lasts at least six months in the fridge. The shelf life is not so different.
If yeast scares you, you may want to check out some of the proponents of the no-knead bread technique.
For a lot more information on this subject, there is a recent publication that covers all sorts of leavening agents.
No, you shouldn't be adding butter at this stage, it will take too much kneading and undo the rising.
Whatever it is you were baking, just go ahead and bake it this way. There are many breads which don't use any fat at all. The taste will be different than with butter, but it will still be a good bread.
Best Answer
You can certainly make pizza dough with baking powder. However, adapting a yeast recipe to baking powder will be very difficult. I would suggest searching for a pizza dough recipe that already uses baking powder rather than trying to adapt your recipe.