I just spoke to someone who has an earthstone pizza oven in their kitchen.
They are thrilled with it. Although it is not going to be part of my kitchen renovation, I just wanted to mention it incase there is anyone else who is looking to install a wood burning oven in their house. The person I spoke to said they did not get a hood but instead had a chimney built for it.
I am in the same boat but have found plenty of options for putting one in our new kitchen. Earthstone is one as are half a dozen other companies (Mugniani, Wood Stone, etc.). The main requirement is to have an air gap around it that would isolate combustable material (wood kitchen cabinets/drywall). This is usually 1 to 3 inches depending on the company. Once you follow this, then every one of these can be installed in residential kitchen. The only hitch I have seen is gas units from Wood Stone which requires a blower exhaust, costing $2K on top of their already expensive units.
The bigger issue is size. These things are huge! 50"+ radius for medium size ovens which are typically recommended for this application. We are still in design phases of figuring out how to fit it in a way that doesn't take over the whole kitchen given occasional use it gets as opposed to other things in the kitchen.
Cost is a major issue too. Mid-size ovens retail for $6K and then you have to have them insalled. And covered with tile or whatever, should you go that route. My quick estimate is that it is going to run you $10K installed. That makes it the most expensive appliance in the kitchen.
Despite the obstacles, we are going to get one. We have been making pizza in a normal oven with a stone and it just doesn't get hot enough to do the job. As to using the BBQ, I have tried that too and it kind of works but it is not ideal. Nor is going outside to do it. Having it right in the kitchen is sure convenient.