Is it permissible to always pray at home and avoid the mosque if going to the mosque causes a loss of khushu' (devotional humility) due to the ill-treatment by some worshippers after a fabricated and inappropriate video spread?

The truth of the matter must be made clear to people, and it must be shown that the video clip is fabricated and a lie, in order to dispel doubts. If you are innocent, Allah will reveal your innocence, just as He did in the incident of the Slander (Al-Ifk). The solution is not to stop praying in the mosque; rather, that could lead to cementing negative suspicions about you and to future negligence in prayer. If you pray at home, there is no sin upon you, provided you are keen to perform it in congregation with your family, for prayer is a salvation and a refuge for the believer.