Is one considered an apostate if he believes Mu'awiyah to be a hypocrite, and then it becomes clear to him that the hadiths he relied upon for his belief are weak or fabricated?

What you mentioned has no bearing on the issue of disbelief (kufr); at most, you might inquire about whether it constitutes a sin or not. Since you adopted your belief based on hadiths you read, and you thought they were from the words of the Prophet, peace and blessings be upon him, we do not see any sin upon you for your belief. However, you are to be blamed for acquiring knowledge from those who are not qualified and for exposing yourself to tribulations by reading on the websites of innovators and those who are astray.