What is the ruling on printing promotional flyers that contain the name of Allah, the Mighty and Exalted, knowing that they may be discarded and trampled underfoot?

As long as you believe that most people will throw papers in the street, your duty is not to write the name of Allah the Almighty on them. The Sharia considers consequences and blocks the means [to evil], and if it is known that most people will respect the sanctity of what has the name of majesty written on it, then there is no harm in writing it.