I don't see any inherent problem with being both a Christian and a fallibilist.
Quote:"The continually progressive change to which the meaning of words is subject, the want of a universal language which renders translation necessary, the errors to which translations are again subject, the mistakes of copyists and printers, together with the possibility of willful alteration, are of themselves evidences that the human language, whether in speech or in print, cannot be the vehicle of the Word of God. The Word of God exists in something else."
— Thomas Paine, The Age Of Reason