It's not a questions of whether they are allowed to.
I'll ask my question yet again. If the builders of this mosque care as much about harmony and mutual respect and understanding as they claim, why do they refuse to do something that would calm the situation such as move the mosque site? It's about motivations Nitrogen.
Maybe it;s just close to where they all work, and they'd like a nearby place to pray 7983598324 times a day?
Maybe the fire escapes look good to them?
Christians often place missions in areas hostile to Christianity, in order to spread the word of God. I know there are Christian missionaries in Saudi Arabia and Pakistan, for instance. I they get caught, they can be executed. Since the Saudis obviously don't want Christians there, why do the missionaries go?
They go because they can have the most impact where they are hated. They can hopefully show people that hate them that they aren't all bad; it's great bang for their buck.
If I was a Muslim wanting to make a difference, I'd want to place my mosque/mission/whatever in that area PRECISELY because people hate me so much. I'd hope, through my deeds, I could show that my religion and my people are nowhere near as bad as others think they are. I'm betting there's an equivalent to Matthew 28:19 ("Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit") in the Koran.
Does your church do missionary work? Would you go to Saudi Arabia to spread the word, if called to?