It seems likely one of the problems is that the cast iron is acting like a heat sink, drawing heat away from the crust instead of letting it bake. Having less moisture, as other answers mentioned, seems like it should help a bit since the heat spent to evaporate the moisture is not spent cooking - but I had a few other thoughts to offer.
You could try pre-baking the crust a bit. Just, stick it in the oven for a bit in the step between pressing the dough into the pan, and adding the toppings afterwards. It will let the crust bake on its own a little bit, and heat up the pan a little bit, so it can bake more even after the toppings are added - so you don't necessarily have to bake it until all the way done, especially if the edges might overcook that way, just enough to get the pan warmed up and the crust par-cooked (like pre-baking a pie crust).
You could try just pre-heating the pan a bit. Stick the cast iron in the oven before pressing the dough into it. You don't need it sizzling hot - that might make it hard to add the dough without scorching your fingers... but if it's heated enough to still be warm when put back into the oven, the pan can heat back up in the oven within a reasonable amount of time, it will let the crust bake, without the cast iron drawing heat away from it.
You could certainly set the pan (with or without dough) on the stove for a few minutes, to specifically heat up the bottom of the crust. This would help prevent the scenario, if you were pre-baking the crust in the oven, where the edges might overcook - since the heat is only applied to the bottom. You could use the stove to pre-heat the pan (without dough), to par-cook the bottom of the crust (with dough), or even set the whole pizza (with toppings) on top of the stove to start cooking from the bottom up, to contrast to the oven's tendency to cook top down (especially if the cast iron is cooling the bottom) - or to pre-warm the pizza to avoid thermal shock to your pizza stone, if you're putting the whole pan on top of it.
Depending on kitchen space and layout, it might even be convenient to leave the pan on the stove-top while you are layering the pizza (perhaps turned on low the whole time, or turned on at some specific point to preheat the bottom), so that pan and the ingredients are slowly warming up the whole time, and in the oven will quickly raise to temperature to start cooking .
Alternatively, you could proceed as you normally do, and just loosely tent the pizza with foil, to let it cook longer in the oven without overcooking or scorching the top of the pizza. You would want to make sure it's loose enough for moisture to escape, or it might steam your pizza toppings or leave the whole pizza too wet. This would probably be the smallest change to make in your recipe.