I say, with Allah's guidance: If the imam fears missing the funeral prayer and begins it with tayammum, it is not permissible; because someone like him should be waited for. If they perform the prayer, he has the right to repeat it. See: Radd al-Muhtar 1: 161, and Sharh Ibn Malik Q13/A. It was authenticated by the author of Al-Hidayah 1: 27, Al-Khaniyah 1: 63, and Al-Nasafi in Al-Kafi. And Allah knows best.