Is it permissible for someone with weakness in Tajwid to lead us in Tarawih prayer, given that the others are not committed?

There is a difference of opinion among scholars regarding the imamate of a fasiq (transgressor). The Hanbalis invalidated it, while the majority disliked it. A just and upright person should lead Muslims in prayer. If your imam is committed and upright, let him continue in the imamate. Weak tajweed does not matter as long as he pronounces the letters correctly from their points of articulation (which is obligatory). As for beautifying the recitation, such as prolongation (madd) and nasalization (ghunnah), it is recommended. Pray behind him and advise him to learn tajweed.