Is it permissible for me to stop praying in a mosque where the Imam does not recite Al-Fatihah well, and to pray in another mosque, knowing that I recite Al-Fatihah well and fear showing off if I lead the people in prayer?

The general rule is that there is no harm in going to a distant mosque, especially if the imam of the nearby mosque does not recite well. However, a Muslim should not abandon the imamate if he knows that the one who will lead in his absence does not recite Al-Fatihah well. In fact, it may become obligatory for him to lead the people if no one else is available who recites well, and he knows that his successor errs in Al-Fatihah with an error that invalidates the prayer. The fear of showing off (riya') is praiseworthy, but it should not be a barrier to doing what has become obligatory for a Muslim. And shortening the prayer, out of consideration for the people's condition, is مطلوب (desirable/required), provided that the pillars and obligations of the prayer are fulfilled, including tranquility (ṭumaʾnīnah).