Is it permissible to perform Salat al-Hajah (Prayer of Need) with twelve rak'ahs, reciting Al-Fatihah and Ayat al-Kursi during prostration, despite the prohibition of reciting the Qur'an while prostrating?

The narration about performing twelve rak'ahs for the fulfillment of a need is fabricated (mawdu'), so it should not be acted upon. Rather, it is preferable to limit oneself to the prayer for need that is authentically reported in the Sunnah (definition).

It is authentically established that reading the Qur'an during bowing (ruku') or prostration (sujud (definition)) is forbidden, based on the saying of the Prophet (peace be upon him): "Indeed, I have been forbidden from reciting the Qur'an while bowing or prostrating. As for bowing, glorify your Lord Almighty therein. And as for prostration, strive hard in supplication, for it is most likely that your supplication will be answered."