With the amount of dough you are trying to make (4 1/4 pounds), you might want to try reversing the process. That is, instead of adding the liquid to the dry ingredients, do the reverse. I would first pour the water into the KitchenAide bowl, and stir or whisk in the salt until it dissolves in the water. That will take about 30 seconds to a minute. I would then combine the flour and the yeast (IDY) in a separate bowl, and gradually add the flour/yeast mixture to the water, a few tablespoons at a time, and combine using either the dough hook or the paddle, whichever is more convenient and works better for you. I sometimes just use a wooden spoon to get the process going and then switch over to the machine.
As you add the flour and it combines with the water, you will get a pretty good idea as to whether you have too much or too little flour in relation to the amount of water. You won't know for sure until you add the oil and knead that in. So I would hold back a little on the flour until the oil has been taken up into the dough to see whether you need to adjust the final amount of flour and/or water. Since I only make an amount of dough for one pizza at a time (all the Lehmann recipes are specified for single pizzas), it takes about a minute or two at most to work in the oil. In your case, it will take longer because you are making a larger quantity of dough. Once the oil has been incorporated into the dough, then you can knead the dough for whatever time it takes to produce a nice, smooth and elastic dough ball. The time it takes to do this will depend primarily on the speed of your mixer. And, with the amount of dough you are making, you can expect a longer knead time than if you were making dough for one pizza. So it's more important to pay attention to the condition of the dough rather than the time. Once the dough comes together and is smooth and elastic and not wet or dry, then you can stop the kneading--whether you were doing all the kneading at speed 1 or speed 2 or 3. I like to finish the process with about a minute of hand kneading to confirm that the dough is at it should be, and to shape the dough into a ball or whatever other shape I want prior to putting the dough into the refrigerator.
If, in the process of doing the above, you discover that you accidentally used too much water, as you indicated in your original post, then you can adjust for the error by adding more flour. If it is only a bit more flour, there is no need to adjust the other ingredients (yeast or salt). Otherwise, you may want to add and knead in a bit more yeast and salt to keep everything in balance.
If you find that the above approach doesn't solve the problem, then you can always resort to making two smaller batches instead of one large batch. Four and a quarter pounds is still a fair amount of dough to knead at one time, even in as good a machine as you have.
Please keep us posted, since I am sure others will benefit from your experience.