Lots of people cook bread or pizza on a grill.
However, the idea of smoking in my mind generally means cooking it slow at low temps.
Many breads cook better fast at high temps IMO.
When my buddy and I do Pizzas, we are talking about 500 degrees on a stone and they are done in a matter of minutes.
I actually plan on the next grill I buy being capable of doing at least 2 pizzas at a time.
However, if you want that smoked flavor, I would say to either stick with the flower or if you do the dough try it during the rising process.
I am no expert in smoking or baking and and although my buddy is a hell of a baker, he is just learning smoking but he agrees.
(BTW, he has over 20 years working in kitchens)
I think our next pizza we might try smoking the dough.
If so, I will let you all know.