At least for the setting of my DW game, Jewel of the Desert, you could probably get away with a precooked-crust style thing as an initial effort, since things like gyros and lamacun/lamajun already exist. Might use a thick yogurt instead of a soft cheese on top, for example, but otherwise would be very similar--savory things like peppers, mushrooms, onions, etc. with or without minced meat, spread over a plate-sized flatbread and baked quickly at high temperature.
And on the question of what categories "pizza" belongs to, I default to my usual answer:
Is it useful to call a pizza a sandwich (or any other similar thing)?
"Sandwich" (and, for that matter, "pizza") are terms invented by human beings, without any systematic categorization. We made them up in order to help us talk about foods. Systematic categorization is only useful if it actually, in practice, makes it easier to talk about whatever is being categorized. Perfectly systematic terms are not necessarily the best terms. If you're genuinely inventing new terms from whole cloth, it absolutely helps to be clear, consistent, and systematic because everyone who hears them is learning them for the first time. But if you're trying to change language that already exists, I'm afraid you need more than just "but but but it's logically consistent".
Pizza is pizza. Further categorization is generally not useful.