Is it true that whoever doubts having fallen into disbelief in the past, or considers it possible that a statement or belief of disbelief may have emanated from them, must immediately utter the Two Testimonies as a precaution, whether the doubt is weak or strong? And is this ruling mentioned in the books of the Hanafis and others, and that whoever denies it disbelieves?

He whose Islam has been established with certainty does not exit from it by doubt, and the default is non-action. Therefore, he is not required to utter the Two Testimonies, and obliging him to do so opens the door to waswasa (whispers/obsessive thoughts). There is no validity to what your interlocutor mentioned, that not enjoining the Two Testimonies means contentment with disbelief, for this is confusion. One must be wary of those who treat matters of faith and disbelief like mathematical equations. The basis for not enjoining the Two Testimonies is that obligation requires evidence, and this Muslim's state of Islam is presumed to continue; the emergent doubt about him is not to be heeded. Unless apostasy is proven, he is not commanded to utter the Two Testimonies. The repentance of an apostate and every disbeliever is his embracing Islam by bearing witness that there is no god but Allah and that Muhammad is the Messenger of Allah, except for one whose disbelief was due to denying an obligation, or a prohibition, or a permissibility, or a prophet, or a book, or the message of Muhammad (peace be upon him). His repentance, along with the Two Testimonies, is his affirmation of what he denied. The most sound opinion among the Hanafis is that being content with the disbelief of another does not constitute disbelief. It is obligatory to beware of waswasa, for what was mentioned in the question is nothing more than opening the door to obsessive thoughts.