Does not speaking to an abusive brother fall under the prohibition of "those who harbor resentment towards each other" mentioned in the Hadith of the Messenger of Allah, peace and blessings be upon him, knowing that one fasts on Mondays and Thursdays?

What your brother does may be a heavy joke or excessive sensitivity, so it is advisable to be patient and overlook it unless you are certain of an intention to harm. If harm is confirmed, then there is no sin in not speaking to him to avoid his oppression, for his action is an offense and an unlawful severance of ties, as evidenced by the Almighty's saying: ﴿So would you, perhaps, if you turned away, cause corruption in the land and sever your ties of kinship? Those are the ones whom Allah has cursed, so He made them deaf and blinded their vision.﴾ [Muhammad: 22-23], and the Hadith of kinship. There is no sin upon you for not forgiving due to contention if keeping away from him is to avoid his evil and oppression, in accordance with the Almighty's saying: ﴿The way is only against those who oppress people and transgress in the land without right. Those will have a painful punishment.﴾ [Ash-Shura: 42]. It is advisable to be patient and remember Allah and forgive, for the Almighty's saying: ﴿But whoever is patient and forgives - indeed, that is of the matters [requiring] determination.﴾ [Ash-Shura: 43].