What is the ruling on my action, which aims to prevent my brother from marrying a girl whom my mother does not approve of, and is my brother allowed to marry someone whom my mother does not approve of, knowing that their relationship began in an unlawful manner (the internet)?

There is no religious impediment to trying to dissuade the girl from marrying the brother to avoid problems. The mother is not permitted to object to her son's marriage without a legitimate religious reason, and the brother is not permitted to marry the girl if his mother's refusal is religiously valid. If the girl is righteous, the brother should try to convince his mother. If she insists on refusing, then obedience to her is obligatory, unless her objection has no religious justification; in that case, he is permitted to marry and should strive to appease her. The initial unlawful acquaintance does not mean the marriage will fail, and repentance to Allah for unlawful conversations is obligatory.