What is the solution to this psychological problem of intentionally vomiting after eating for fear of gaining weight, and is there anything in the Hadith or a Quranic verse that deters from this act?

It is not permissible to intentionally induce vomiting for the purpose of losing weight, as this is Islamically forbidden. It constitutes a crime against oneself, a waste of wealth, and an ingratitude for God's blessings. God tests with abundance to see gratitude and proper utilization of blessings. You must fear God and be grateful for the blessing of sustenance. You must also know that squandering wealth, extravagance, and profligacy are reprehensible and forbidden acts. Remember the state of the Prophet, peace and blessings be upon him, who suffered from a scarcity of food, and consult trustworthy doctors if your condition is an illness.