Is it permissible to say: "O Allah, bestow Your prayers upon Muhammad and his family, and grant them abundant peace, as many times as Your creation, to the extent of Your pleasure, the weight of Your Throne, and the ink of Your words"?

There is no harm in sending blessings upon the Prophet, peace be upon him, using the formula mentioned in the question, because it is a supplication that does not contain any Sharia violation. Sheikh Al-Islam Ibn Taymiyyah is of the view that the reward for this remembrance is one, but it is greater than the reward for mere prayer. Ibn Al-Qayyim is of the view that this "multiplied remembrance" is a greater praise than individual remembrance, because it includes the knowledge of Allah, His transcendence, and His glorification. However, one should not believe it has a special virtue or that it is a confirmed Sunnah or that it is better than the Ibrahimiyya prayer, and it should not be used in prayer.