There are two reasons (that have also been discussed in many other questions)
Baking powder isn't just sodium bicarbonate + acid. It often also contains aluminum compounds that release gas when they are heated. That means they will make bubbles not just when the batter is mixed but also when it is baking.
Baking powder is ph neutral while baking soda is basic. Recipes will often include mostly baking powder with some soda to neutralize acidic ingredients.
See also:
What is the difference between baking soda and baking powder?
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
A quite readable explaination from a supplier of leavening agents : http://www.lallemand.com/BakerYeastNA/eng/PDFs/LBU%20PDF%20FILES/1_12CHEM.PDF
According to them, sodium acid pyrophosphate has indeed a "slow to very slow action, slightly bitter aftertast". But the bicarbonate will produce CO2 when reacting with the acid, regardless of heat.
I wouldn't use ammonia bicarbonate as its "ammonia taste limits [it] use to low-moisture products" : your batter might still contains too much moisture to get rid of the ammonia taste.