That really sound gross, sorry I am not trying to be rude, but that just sounds nasty with a smoke flavor in a crust
I feel would may be better off by trying to use a smoked cheese (maybe a smoked provolone?) or lightly misting the the top of the pizza with a diluted liquid smoke solution before baking.
The wood-fired pies that I have had have not had a smokey flavor except for one where the guy had fuelwood with too high of a moisture content and it was burning poorly and creating a dark pungent smoke, the wood was not even capable of generating enough heat to burn the soot out of the oven dome, so the pizza suffered all the way around.
Or, how about getting a wood-fired oven? $14K will get you an ANSI/UL rated commercial 60" oven from Forno Bravo that is ready to fire and cook in the same day it is delivered. And you can even have a gas back up burner with that too.