I estimate that the flour component of the Buttermilk Bisquick mix is around 5.7 ounces, or about 162 grams. We cannot assume that 1/2-cup of water called for by the instructions on the package is the right amount to use if we are to use the Buttermilk Bisquick mix as a base for a pizza mix. That amount is to make biscuits, not pizza dough. So, I would use an amount of water (noted below) that is more in line with what I would use with a cake flour, around 55%.
To the Buttermilk Biscuit mix, I propose to add 6 ounces (170 grams) of King Arthur Bread flour (KABF). Using the Mixed Mass Percentage Calculator at http://foodsim.unclesalmon.com/,
and assuming 7.9% protein content for the presumed cake flour in the Buttermilk Bisquick mix and 12.7% for the KABF, I calculate that the protein content of the final blend is around 10.3%. That is in the all-purpose flour category. I also believe that that value is higher than the flour component of the Betty Crocker pizza crust mix you used recently to make your mystery pizza. Adding the 6 ounces of KABF to the Buttermilk Bisquick mix will have the effect of reducing the amount of total fat in the blend. It will also reduce the effects of the egg, buttermilk and soy flour in the Buttermilk Bisquick mix. You would have to add more of these ingredients to get to the levels of the Buttermilk Bisquick mix. I don't see a compelling need to add more of these ingredients but if you have them and want to use them, I can give you some percents to use (relative to the weight of the KABF).
I also propose to add more baking powder (Clabber Girl brand), more salt, some sugar as an alternative to dextrose (which I assume you do not have), and some instant dry yeast (IDY). The IDY is intended mainly for crust flavor and possibly some fermentation if such can be achieved in the short dough preparation time. Of course, you could let the dough ferment at room temperature to help achieve more fermentation and get some of the benefits that the longer fermentation would confer on the finished crust. I will leave to you to decide on whether you want to do that. Maybe you want to see if you can make an "instant" dough (just as you did with the Betty Crocker pizza crust mix) and eliminate that step.
Here are the steps to follow to try to transform the Buttermilk Biscuit mix to a pizza crust mix:
Add the contents of the Buttermilk Bisquick mix packet (212 grams/7.48 ounces) to a bowl
Add 6 ounces (170 grams) of KABF
Add 4.59 grams/0.16 ounces of the Clabber Girl baking powder; this comes to 1 1/8 t.
Add 2.55 grams/0.09 ounces salt; this comes to a bit less than 1/2 t.; you will have to adjust if you decide to use a Kosher salt
Add 2.04 grams/0.072 ounces of sugar; this comes to 1/2 t.
Add 2.26 grams/0.08 ounces of IDY; this comes to 3/4 t.
Total weight of "goody bag" contents = 6.4 ounces/181.44 grams
There are two components of the water:
Water 1: 93.5 grams/3.3 ounces (this is in relation to the presumed cake flour in the Buttermilk Bisquick mix)
Water 2: 105.4 grams/3.72 ounces (this is in relation to the KABF)
Total weight of water: 194.9 grams/7.02 ounces
In light of the novel and uncertain nature of this experiment, you may want to initially hold back on some of the water to be sure that all of it is needed. This is where you may have to exercise your professional judgment as to when the dough has the proper consistency and "feel". That consistency and feel may be the same as when you made your mystery pizza. So, all of the water may not be needed, or maybe more will be needed. I suggest that you use a water temperature of around 130 degrees F, or whatever comparable temperature you can achieve out of the tap at market. If possible, I'd like to see some
The total dough batch weight, at least on paper, should be around 592 grams/20.89 ounces. That amount of dough should be just about right to make a single 16" or maybe slightly larger pizza (with a thickness factor of around 0.10), or two roughly 12" pizzas (with a slightly smaller thickness factor value).