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.
The short answer to your question is YES. The extra acid in the ingredients will hamper the second act of the double acting baking powder. The acids are timed/staged for reaction not the baking soda.
The Magic Baking Powder (happens to be in our kitchen, too) is mostly a single acting formula since monocalcium-phosphate is a low temperature acid (with apparently some double acting properties due to generation intermediate step of dicalcium phosphate; per your link). High temperature acid for second acts typically include sodium aluminium sulfate, sodium aluminum phosphate and sodium acid pyrophosphate.
You can try to counteract that by adding a bit of baking soda, but you run the risk of altering the taste and not having it all neuralized.
If you really want to get pedantic, use a pH meter to measure the acidity of your dough. I suspect tasting the dough might give an indication as well. (bitter alkaline, sour acidic)
Another test might be to mix your acidic ingredients in a bowl with some water and start adding measured baking soda until you see no more reaction (bubbles) and use that as a your basic of neutralizing your dough.
All said and done, I agree with SAJ14SAJ that you'll be just fine going with the existing recipe. There should be enough baking soda left to get something out of your double act. I also heard it from a world-class baker that most recipes can be done with only baking soda, let alone baking powder or double acting ones.
Best Answer
You can make yeast free pizza base using baking powder, it will be a bit crumbly, but it works fine. You could use baking soda, but then as you state you need acid to react with it. Using lemon juice is an option but I would recommend against it as you don't know the exact amount of acidity you'll get and your crust will taste lemony which you don't want. Go get some baking powder instead and use about 3/4 tsp of powder per cup of flour.
You'll want to knead your dough to develop the gluten, otherwise the base will just fall apart.