Can a non shomer shabbos Kohen duchan?
If he desecrates Shabbos publicly, it is preferable that he should not duchan. However, if he acends to duchan, he should not be stopped.
The Shulchan Aruch Harav (128:52), the Peri Megadim (52) and the Mishna Berura (134) rule that somebody who publicly violates the Shabbos should not participate in the blessing.
However, Rav Moshe Feinstein (Orach Chaim 1:33) argues that the comparison between violation of Shabbos and idolatry, which is the source of the prohibition (see Peri Chadah 128), is certainly not absolute, as can be derived from the fact that repentance surely aids the Kohen who once desecrated Shabbos (unlike an idolator). After trying to shy away from replying, Rav Moshe rules that the Shabbos violation of Jews of his time does not demonstrate kefirah, as public violation of Shabbos used to do in the past. Similar logic can be applied to Shabbos violation today.
Therefore, although it is preferable that he should not participate in the blessing, if he ascends to ducham, he should not be prevented from doing so.