Is it permissible to perform tayammum and omit the recitation of Al-Fatihah, contenting oneself with tasbeeh and takbeer in prayer, when experiencing extreme hardship in purification with water and the recitation of Al-Fatihah?

If the cause of hardship in ablution (wudu') and reciting Al-Fatihah is the feeling of an impediment, then this is not an excuse that permits switching to tayammum instead of wudu', or to tasbeeh instead of Al-Fatihah. You must fear Allah and do what He has commanded, which is to perform wudu' and recite Al-Fatihah in prayer, and to strive against your inner self (nafs). One only resorts to tayammum if incapable of using water, and one only switches from Al-Fatihah to tasbeeh and tahleel if one does not memorize it and was unable to learn it. What you feel might be an illness or a touch from jinn, in which case seeking treatment is legislated for you.