Personally, I'd also look at the environmental effects of having a pizza oven - I couldn't justify it given we'd cook two or three pizzas for a family of 4 adults. It's also why I got rid of our big BBQ. Unless OP plans on having lots of family round, how often will the oven actually be used properly ?
Our little cast Iron BBQ is more than adequate to cook for 4 adults, which in turn burns less than half what the big BBQ needed to cook for 4. My neighbour has a monsterous metal pipe outside is modern fully CH house, all for his log burner, which he burns all his building waste on. There is no need for a burner - our gas CH and cooking costs just £40 a month !
My opinion I know, but it's becoming a big issue with wood burning appliances, especially in clean air zones (like where I am).