I can see that the prescribed bases made life a lot easier because the OP could just come home and sling toppings on, cook and hey presto! Quick cheap dinner.
Compared to that, having to plan ahead, learn to make bases properly, and freeze batches to have available must seem a massive effort.
But it really really isn't unreasonable to expect the OP or other GF sufferers to do that. The prescription is surely to give people necessary staples, not convenience.
It does just mean switching to a different kind of cooking-in advance, in batches. The more you do it, the easier it gets.
I Googled 'No Knead Gluten Free Bread' and found a lot of results. I picked out this one which sounds easy to do but does include expensive ingredients you'd have to track down. Extra faff, no doubt. But doable with planning.
And this website looks lovely, but again, unusual ingredients mean this isn't an easy throw together option.
And the, ta-daah! courtesy of thekitchn.com, I found this:
**
Yeast-free Pizza Dough
Makes 1 large pizza or 2 individual pizzas
2 cups self-rising flour
1 tablespoon cornstarch
1 teaspoon kosher salt
3/4 teaspoon garlic powder
1/2 cup water plus additional as needed
2 tablespoons olive oil
1 teaspoon honey
Place a pizza stone in the oven and preheat to 500°F.
In a large mixing bowl whisk self-rising flour, cornstarch, salt, and garlic powder together until mixed. In a separate bowl, combine water, olive oil, and honey. Add the liquid ingredients to the dry ingredients and stir until a cohesive ball forms, adding more water a tablespoon at a time if necessary. (The dough should be moist and firm but not sticky.)
Form the dough into a rounded disk and wrap with plastic wrap. Refrigerate for 15-30 minutes. On a lightly floured surface, roll out the dough with a rolling pin to form a perfect circle. Transfer the dough to the hot pizza stone and cook in the oven for 2 minutes.
Remove the dough from the oven. Top with tomato sauce, shredded mozzarella, and additional toppings as desired. Cook until the cheese is melted and the crust is very crisp, another 7-10 minutes. Cool for 3-5 minutes before serving.
**
Got to be worth a try, OP?