Uh, CO is about 28 DA, whereas N2, O2, and CO2 are about 28 Da, 32 Da, and 46 Da respectively. Under standard conditions the density of CO is about 1.25 g*L-1, whereas N2, O2 and CO2 are 1.25 g*L-1, 1.43 g*L-1, and 1.98 g*L-1. So no, carbon monoxide is not more dense than air. Add to that the fact that carbon monoxide produced will be at a higher temperature than the surrounding air and that it will be caught in convection currents moving upward and it is inaccurate to say that the carbon monoxide will sink.
That being said, there is a reason that restaurants use updraft hoods. I would not recommend the two box fan idea, as you could still accumulate significant concentrations of dangerous gasses, CO being only one of them, if your airflow is not properly designed. If you are intent on using gas indoors you should consult an engineer. Remember that your one, two, or three banjo burners are cranking out a lot of gas. Whether it burns up or not it's at the least displacing oxygen and is not something that's worth experimenting with.