Welcome to the BrickWood forums!
First, thank you for being thoughtful about not adding to the solid waste mountain by simply throwing these out. I’m guessing from your post that you already know what you have there, but I’m adding some notes in my response in case someone else is thinking similarly.
Insulating FIre Brick (IFB) is used in kilns and fireplaces. It is lightweight because its component materials are mixed with an organic material that is “burned out” during the process, resulting in a very porous structure. Heat gets trapped in the air spaces, much like it would in other insulators like fiberglass or the mineral fiber blanket recommended for the BrickWood ovens. It cuts easily, which is a help when incorporating it into a structure.
Hard Fire Brick is used in production environments like coke ovens, crematories, and (yes) pizza ovens. It is a dense refractory that has alumina added to the mix. It is designed to refract heat rather than insulate. It is very durable during firing, and can stand up to chemical and mechanical wear. It has excellent resistance to thermal shock.
You use IFB when you are trying to reach a set temperature and you won’t be banging the brick lining or spattering it with chemicals.
You use hard firebrick for superior mechanical resistance and when there is likely to be thermal swings and extremes — like direct flame and pans and peels. Because it is dense, it is a better structural material than IFB (which itself is pretty good).
I’d kind of lean toward don’t apply the IFB as part of your oven. As many BrickWood builders can attest, even the low and medium duty firebrick takes a hot beating when the oven is fired. Adding a material with different properties is going to make your refractory lining (which is what you’re really building with the Barile design) unstable and unpredictable. If it turns out that one or more of the IFB fails, how would you fix it?
There may be other ways you can apply the IFB in a complementary project — there are LOTS of completed project photos on these forums. If there’s enough to line a fireplace, for example, that would be a great application for this material.
Again, I’m glad you’re considering other uses for the material than throwing it out!