That problem is not with the use of “Allah” but with 99% of the world lacking education. In the case of the Muslims in particular, it would do them good to understand that they do not have a monopoly on the faith in Allah.
I actually doubt that the Muslims “don’t know of the existence of Arab Christians” — they live with them, sometime peacefully, sometime not. In Iraq, Syria, Egypt, Arab Palestine Christians are a noticeable minority, and in Lebanon they are close to half the population. These are all Arab countries are they not?
The great majority of Muslims are not Arabs, and even in Arab countries, Christians are a small minority. When a Muslim hears someone use the name Allah, he is thinking of Mohammeds God, and that that person has rejected the polytheism of traditional Christianity.