Could we accept as a Shaliach Tzibur someone who is Shomer Shabbat in public but not fully in private. As a person, he is fully accepted by the community, has the vocal qualities and a good moral standing. Please back up your answer with full references.

ב"ה The easier part of your question is providing the references. However, though to begin with the Shaliach Tzibur should be the most G-d fearing person, defining the person you are describing is more difficult because there are many variables in accepting the information of a cillul Shabbat in private. Also there is what to look into of what was done actually constitutes chillul=desecration of Shabbat, and how to relate to a mechalel Shabbat in our times. There are also community considerations which must be made Therefore, please forgive for leaving this answer for your community Rabbi to deal with. On the other hand in regard to references, please see the following: Shulchan Aruch Orach chaim 53. See Meshiv Milchama vol. II, chapter 107 of Rav Shlomo Goren zt"l. See also Yalkut Yosef, Vol I, Hilchot Shliach tzibbur, of Rav Yitzchak Yosef shlit"a, the current Sefardi Chief Rabbi of Israel. See also Mareh Habazak, Vol. III, Chapter 6. Chag Same'ach.