Is shaking hands with the congregants and the Imam reciting Surah Al-'Asr after Fajr prayer an innovation (bid'ah) or a Sunnah, should they be denounced for it, and is the evidence they rely upon sound?

The actions of your group after Fajr prayer, specifically the imam's consistent recitation of Surah Al-Asr, shaking hands, and supplicating with "Allahumma ij'alna minhum" (O Allah, make us among them), are not from the Sunnah. It has not been reported that the Prophet (peace be upon him) used to recite Surah Al-Asr to his companions after Fajr, nor that they would shake hands after Fajr.

The reported tradition from the companions regarding the recitation of Surah Al-Asr upon meeting does not imply the imam's recitation, nor does it specify a time after Fajr.

Therefore, making this a regular habit after Fajr is considered a specification for which there is no evidence, and it makes something that is not Sunnah into Sunnah. This is how many innovations (bid'ahs) originate.

Ibn Mas'ud (may Allah be pleased with him) said concerning the tribulation from which innovations arise: "How will you be when you are overtaken by a tribulation in which the old man grows decrepit, the young one grows up, and people adopt it as a Sunnah? Then, when it is changed, they say, 'The Sunnah has been changed'?"