Is my prayer valid if I recite the Basmalah at the beginning of Al-Fatiha, believing that it is not one of its verses, and then sometimes repeat Al-Fatiha without the Basmalah due to lack of devotion, knowing that I recited it at the beginning?

The opinion of the majority is that the Basmala (In the Name of Allah) is not part of Al-Fatiha, whereas Al-Shafi'i holds that it is a part of it. Reciting it is legislated to avoid disagreement. As for repeating Al-Fatiha, it is better not to do so to avoid the disagreement of those who invalidate the prayer by it, even though the preponderant opinion is that it does not invalidate it. Hanbali jurists disliked repeating Al-Fatiha, and this was justified by avoiding the disagreement of those who invalidated it. العلامة Al-Uthaymeen permitted repeating Al-Fatiha if it was to achieve a legitimate religious purpose, such as someone who forgot and recited it silently in a place where audible recitation is prescribed, or wanted to repeat the recitation to attain humility in prayer, warning against repeating it if it leads to opening the door to obsessive whispering (waswas).