Since, unlike Christians, Muslims do not believe in a Trinity CH”V, the Muslim “Allah” should refers to what we refer to as “G-d”. Given this, it makes sense to use a Muslim term when speaking to Muslims.
I also don’t see the gross blasphemy in calling Yerushalayim “Al Quds”, when speaking to Muslims, especially when you would refer to it as “Jerusalem” if speaking to any non-Jew. Al Quds simply means “HaKodesh” just as “Jerusalem” really means “Yerushalayim”, which is the “Ir HaKodesh”.
(To be clear, yes, it is *really* “Yerushalayim” as that’s Hashem’s name for it. Interestingly, it seems the city was of no interest to Muslims, as is well known, until one of the Pashas conveniently decided a few hundred years ago that Al Aqsa refers to a new Mosque on Har Habayis. But that’s irrelevant here.)