There are not a lot of people in our community who can serve as Shaliach Tzibur on Shabbat. We want Shlichei Tzibur who can uplift the kehillah with singing and proper nusach. Under these circumstances, it is permissible for an aveil to serve Shaliach Tzibur?
It is permitted. Even Rema (YD 376:4) who rules that an aveil not to be a shaliach tzibur on Shabbat and Yom Tov
אבל לא נהגו להתפלל בשבת ויום טוב, אף על פי שאין איסור בדבר
makes it clear that it is a minhag and not technically an issur. And poskim thus make exceptions when necessary.