I agree with you, but with one detail. In the early days of Islam the Muslims were all united during their expansion which allowed a wider diffusion of their religion.By that logic, the Middle East should never have become majority-Muslim in the first place, since Christianity already had an organised religion and structure, and the Muslim conquerors didn't conduct a brutal campaign of religious genocide against their Christian subjects.
Anyway, the best way, IMHO, would be to do what the Caliphate did, but in reverse -- i.e., give Christians various legal and tax privileges, support Christian church-building and missionary activities whilst restricting similar activities from Muslims, and make it so that certain high-status positions are only open to Christians. People would start converting to Christianity as a result. Given enough time, there'd be enough converts for Christianity to be mainstream and common once more, as the OP requires; given even more time, Christianity would become the majority religion in the area.
I don't see Europeans uniting so easily even for religious reasons. Moreover they will have to make a lot of effort to push the population to convert without provoking revolts.
What you are proposing is possible, but it requires an overpowering and united Europe.