Does acidity negate double-acting 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.