Is it necessary to fulfill the aforementioned vow (refraining from backbiting, coupled with giving two dirhams in charity upon violation) after expiating for its breach, without renewing it, especially since the breach occurred four separate times?

Abandoning sin is a religious obligation, and it becomes more emphatic if a vow or an oath is made concerning it. It is not permissible to break such a vow or oath. Whoever says, "By the covenant of Allah, I will not do such and such," and then does it, must offer expiation for an oath (kaffarat yamin). What you stipulated about giving charity if you backbite someone is a vow of obstinacy (nadhr al-lajaj), and most scholars hold that it requires the expiation for an oath. The oath is dissolved as soon as it is broken and expiation is made, unless the one who swore it intended to break it repeatedly.