Is it permissible to write a research paper discussing the beliefs of Christians for the purpose of inviting an irreligious friend to Islam, knowing that refusing to do so might sever ties with him?

If the researcher comments on what contradicts Islamic beliefs and explains its falsehood, then there is no harm in publishing it. However, if it is merely a narration of the words of a Christian writer without explaining its falsehood, then it is not permissible to publish it. It is not befitting for a Muslim to write words that contain insults to Allah Almighty. The excuse that the goal is to win his heart does not permit the writing of shirk (polytheism). Allah Almighty forbade His Prophet from expelling the poor Companions out of desire for the polytheists to embrace Islam, and He rebuked him for turning away from Ibn Umm Maktum in order to dedicate himself to the leaders of Quraish. It was your duty to inform him of the ruling on his action so that you would not fall into sin.