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.
This is not really an answer,
but rather a report on an experiment.
After the discussion here I got very curious and wanted to compare what I would call a "yeast cake" (even though this is against the traditional definition, but the texture is more or less that of a spongy cake/quick bread) to the "same" cake made with baking powder.
To perform the comparison I divided all ingredients in two and the only difference between them was the leavening.
I tried to keep it really simple, and threw together some ingredients that I had in the house (using a simple quick-bread/muffin method).
Just so you have the feeling of what the batter was like, I will add my experimental
materials:
For each cake I used:
2 ¼ cups soy milk
¾ cup baking margarine
3 cups all-purpose flour (type 550)
2 cups raw sugar
¼ cup cornstarch
½ tsp ground vanilla pods
Then for one (A) I added:
1 ½ tsp dry yeast
2 tbsp sugar
and for the other one (B) I used:
1 tsp baking powder
½ tsp baking soda
2 tbsp cider vinegar
For both I warmed up the milk and margarine just enough that it melted and added the vanilla, then let it cool down.
For A I then added the extra 2 tbsp of sugar and the yeast, and let the yeast get activated and work for an hour - the soy milk curdled and completely separated. When I made B I added the vinegar to the milk mixture and mixed - the soy milk got thicker and curdled a little. The baking powder and baking soda I added to the dry ingredients.
After waiting for an hour for A to do its thing (of course B was ready to put in the oven instantly), I poured the milk mixture into the combined dry ingredients and gently stirred, just enough for the components to combine. I poured the batters into oiled and floured pans and baked each cake on its own in a preheated oven for 40 minutes (this is when both passed the skewer test) at 350 F.
Results:
Both cakes rose to 2.5 times the batter height/level. However, B rose more evenly, whereas A rose a bit less towards the edge of the pan. B smelled like a normal cake/quick-bread, but A filled the house with a "doughnuty" smell (the most scientific term that can be used here would probably be: super yummy). When cooled and cut, both had a really (equally) nice bouncy, fluffy, slightly crumbly spongy texture. A had a bit more and larger trapped bubbles, see figures 1 and 2.
One of my taste testers (or is it test tasters) is quite sensitive to baking powder, and said B tasted a bit too much like baking powder. Everyone agreed that A tested a bit more complex. Both cakes would probably have been better with some nuts or chocolate or fruit, but I avoided these in order to have less complex experimental conditions, to make the cakes easier to compare.
Fig. 1: Side-by-side comparison of the yeast leavened "cake" (left) and the baking powder leavened cake (right).
Fig. 2: Zoomed in image of the yeast leavened "cake" (left) and the baking powder leavened cake (right).
The experiment was performed using an electric oven as I don't have a bread machine. A comparison between this two would be interesting as well.
Best Answer
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.