What is the ruling on a Sunni Muslim man and his non-Sunni life partner having their marriage contract written by a non-Sunni sheikh?

The answer referred to the ruling on marrying someone who differs in belief, and on someone who differs in belief acting as an agent for solemnizing the marriage contract. If their innovation (bid'ah) is considered kufr (disbelief), then it is not permissible to appoint them as an agent. Otherwise, it is permissible. However, if the intention is their role as an instructor for the صيغة الإيجاب والقبول (formula of offer and acceptance) (the ma'dhun or officiant), then it does not matter if they are a disbeliever or a fasiq (transgressor).