All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Reciting Surat Al-Fatihah-before or after supplication- with the intention to invoke Allah for acceptance is lawful for two reasons: First: Invoking Allah by reciting Quran is invoking by an attribute of his, and this is lawful by consensus of Muslim scholars.
Second: Invoking Allah by reciting Surat al-Fatihah is among the good deeds, and this is also lawful by consensus of scholars. Moreover, choosing Surat Al-Fatihah in particular is acceptable from the perspective of Sharia for it is "The mother of the Book" in which all meanings of the Noble Quran are embodied. And Allah Knows Best.