What is the solution for someone afflicted with an annoying involuntary movement near the mouth that has led to social anxiety, negligence in prayer, and fear of ruqyah (exorcism), and can they despair of God's mercy because of that?

You must thank Allah for the blessing of commitment, and for patience during affliction, and look to those who are less fortunate to appreciate Allah’s blessings. Calamities require patience and seeking reward to attain recompense, otherwise there will be despair and discontent, with the calamity and its sin remaining. There is no need to fear Ruqyah (definition) Shariah. Overcome this stage by stopping thinking about the matter, ignoring the opinions of others, repenting to Allah, striving in obedience to Him and thanking Him, and striving in work and showing kindness to people.