If I understand your post correctly, what you are trying to do is use two preferments, one based on using commercial yeast (IDY) and the other being a natural preferment. Plus you are using some extra IDY in the final mix when the commercial preferment and the natural preferment all come together.
Using two preferments in the same dough is a bit unorthodox but not unprecedented. As a matter of fact, that is essentially what Anthony Mangieri of Una Pizza Napoletana does. In his case, he uses a combination of old dough and a natural preferment. I am pretty certain that his old dough is taken from a prior day’s production and is also a natural preferment, not one based on using commercial yeast. I think Anthony uses the old dough method because it is an approach that he feels comfortable using, having been his original approach before he started to use the second preferment, which he formed by using wild NYC yeast and flour and water in his pizzeria.
What you did by combining half of the flour and half of the water plus yeast (IDY) seems to come closest to a sponge technique, that is, one using a portion of the water, enough flour to produce a medium hydration preferment dough in the range of about 60-63% (in your case, 60%), plus some yeast. There is nothing wrong per se with using this approach but the sponge process has to be designed properly, especially if you have a finite amount of IDY available to use. For example, if you use 50% of the flour for the preferment, as you did, and would like to have a room temperature fermentation time of say, 8 hours (assuming a water temperature of 60 degrees F and a room temperature in the range of 80-85 degrees F), you will need about 0.23% IDY by weight of the preferment flour, or 0.0184 ounces IDY. 0.0184 ounces of IDY is more than the total weight of IDY (0.01 ounces) used in the Raquel dough formulation. This means that you would have to use less yeast and a longer room temperature fermentation, or use warmer water, or a warmer room temperature, or some combination of these factors. You could also reformulate the preferment to use less flour, and hence less yeast, but still stay within the ranges normally used for sponges.
It’s hard to say what will work best for you without further experimentation, but you could try using all of the formula IDY in the sponge, which is very common. From this point forward, the fermentation time will depend on your water temperature and the room temperature in your kitchen. Until you get things right and note how long everything takes, you would have to watch the preferment to determine when it reaches the maturation/ripening point signaling that the preferment is ready to be used. Otherwise, you could alter the characteristics of the final dough and the finished crust. I would add the poolish preferment to the final mix at the same time as you add the natural preferment. From that point on I think you should be able to use the rest of the Raquel dough processing steps.
Another approach you could consider is to make your own old dough from scratch, using flour, water, yeast, and salt. However, it too, would have to be designed to make use of the available IDY and otherwise comport with the usual rules that govern the formation and use of the old dough. Otherwise, as with a sponge, the final results will be unpredictable and possibly produce mediocre results or create other problems with the final dough. Remember, preferments go through the same general processes as regular doughs, so quantities and times and temperatures are very important. Preferments are math and science at their very best. You just can’t toss a bunch of ingredients together and get the results you are hoping for. Professional bakers know this and experiment all the time with their preferment formulations and characteristics to find what will work best for them, consistent with the need to be able to produce bread on a commercial basis. Remember also that when you are playing around with preferment formulations used for bread baking, the finished results can take on flavors and other characteristics normally associated with breads. For example, when I tried using a biga preferment to make pizza dough, the finished crust tasted a lot like a baguette. It was nice but not quite what I expected. You might discover that you don't like the "new" flavors.