Is it permissible for a man to marry a girl he secretly loves, with her consent and approval, and with a sheikh acting as her guardian instead of her rejecting father, especially since this man is already married and wants to marry this girl secretly while keeping his first wife, and which madhhab permits this?

The guardian (wali) is a condition for the validity of marriage according to the majority of jurists. Justice is not a condition for his guardianship; thus, the unrighteous guardian does not lose his guardianship. If a girl asks her guardian to marry her to a suitable man, he is obliged to marry her. Otherwise, he is considered to be preventing her from marriage (ʿāḍil), and the guardianship transfers from him. She then has the right to take her case to the judge to marry her off. As for concealing the marriage from the first wife, there is no harm in it, as announcing the marriage is not a condition.