Is it permissible to say: "O Allah, send prayers upon our Prophet Muhammad—prayers by which distress is relieved, knots are untied, and doors of goodness with You are opened for us"?

There is no harm in sending blessings upon the Prophet, peace and blessings be upon him, using the aforementioned formula, as long as the one praying does not believe in its exclusivity, since there is no Sharia violation in it. However, sending blessings upon him using the transmitted (Ma'thur) formulas is better and more blessed, as stated in the Hadith of Abu Mas'ud al-Ansari: "O Allah, send Your blessings upon Muhammad and upon the family of Muhammad, as You sent Your blessings upon the family of Ibrahim. And send Your barakah upon Muhammad and upon the family of Muhammad, as You sent Your barakah upon the family of Ibrahim in all the worlds. Verily, You are Praiseworthy, Most Glorious."