All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
There are some facts that should be stressed first of all: the cross is a falsehood and a ritual that is peculiar to non-Muslims, the Christian religion was distorted and the Injeel (Gospel) that exists today contains many lies. This is in addition to the fact that Christianity is one of the divine messages that were abrogated by Islam. These facts are unchangeable in any way.
As for the demons fearing the cross or feeling pain at reciting something of the Bible, who can confirm that? And supposing that it is true, then what will this mean? It is falsehood anyway and this, if it actually happens, does not change anything of its ruling. The sorcerer may have an effect on the demons and may get them out from the body in spite of his disbelief and misguidance. The same applies to the case in question.
A person being non-Muslim does not mean that the devil cannot be taken out of his body if he is obsessed with it. As for the priests, they may get it out through the ways they know.
Finally, take note that it is permissible in Islam to make Ruqyah (healing with the Quran) for the non-Muslim and recite something of the Quran on him as it was reported in the story of Abu Sa‘eed Al-Khudri when he made Ruqyah for a non-Muslim man by reciting Soorah Al-Faatihah on him and the Prophet, sallallaahu ‘alayhi wa sallam, approved that as reported in Al-Bukhari and Muslim. For benefit, please refer to Fatwa 88454.
Allaah knows best.