What is the ruling on a married man marrying another woman by her appointing herself (as his wife) without witnesses, knowing that he follows some jurisprudential schools of thought that permit this? Is his marriage valid?

Witnesses are a condition for the validity of marriage according to the majority of scholars. However, the Malikis consider it a condition for the validity of consummation and building a home with the woman, not for the validity of the contract itself. The Shiites and some Sunni jurists disagreed, considering witnessing to be merely recommended. What has been attributed to the Malikis regarding the permissibility of "talfiq" (combining different scholarly opinions) is incorrect, because following dispensations (rukhs) is unanimously forbidden by Islamic law, as it leads to the abrogation of religious obligations and takes the accountable individual away from piety and towards following whims. Therefore, "talfiq" in marriage is invalid, and your fear of fornication or being alone with a woman does not permit you to engage in this forbidden matter.