Can one lechatchila appoint a shaliach tzibur who is known to publicly transgress a biblical prohibition?
What if there is/isn't another person who can lead the prayers instead?
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Rabbi Moshe Shternbuch discusses a quandary that was asked of him several decades ago in South Africa. The synagogue had to choose between two individuals to be their cantor for high holidays.
Mister A was a relatively traditional Jew, but he drove to synagogue on Shabbos.
Mister B was a Kohen who kept shabbos, kosher, you name it. Only one problem -- he fell in love with a woman who wasn't Jewish. She'd convert, but as he's a kohen he couldn't marry her any how. So he was married to a non-Jewish woman.
If I recall correctly, Rabbi Shternbuch ruled that Mister B was preferable, with many others opining it should be Mister A. But yes, rabbis have had to deal with such questions.
It's reasonable for a community to have standards, and if someone's public behavior makes him clearly not this community's "emissary" (which is a cantor's real job), well then so be it. But assuming your congregations doesn't particularly care, then to the best of my understanding, so long as he' obligated to pray, then he can pray for you too. (If I recall it's the Kitzur Shulchan Aruch who lists a whole long line of qualifications for the High Holiday cantor job. Then says -- but the only thing that's absolutely required is "adult Jewish male.") We can discuss what to do with a Jew who knowingly walked away from all of observance, of publicly defies its theology, or intentionally and publicly flouts the observance of shabbos (shabbos is the exception that proves the rule where public violation could put someone in a very different halachic category). But other than that? I'm not aware of a source that says he can't lead prayers.
Rabbi Moshe Feinstein discusses a kosher slaughterer who, if I recall correctly, had applied for employment (or was employed?) at a heterodox synagogue that employed a microphone on shabbos. Rabbi Feinstein ruled that the degree of public Shabbos violation was not enough to render his slaughter totally not-kosher; however we would not trust such a person's say-so that his food is kosher. Hence his actions are ritually valid, but they practically needed more oversight. Presumably the same would apply here, if the meat he slaughters (assuming we saw him do it right) is kosher, then he could pray and others could listen to fulfil their obligations.