A Chazan Davening a different Nussach than the congregation:[1]

From the letter of the law, it is permitted for a Chazan to Daven for the Amud in his Nusach even if it is a different Nussach than the Shul and congregation.[2] Nevertheless, if this matter will lead to strife or dispute, he is to Daven the Nussach of the congregation.[3] Practically, the public directive of the Rebbe Rayatz in such a case is for the Chazan to Daven the Nussach of the congregation.[4] However, nonetheless, the Chazan is to Daven his own private Nussach by the silent Shemoneh Esrei.[5]

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[1] See Poskim in Nitei Gavriel 54/4; Shulchan Menachem 1/171-174; Piskiey Teshuvos 68/4

[2] Piskeiy Dinim Tzemach Tzedek 236 regarding that Davening a different Nussach than the congregation does not involve Lo Sisgodedu [He however does not discuss the scenario of a Chazan]; Maaseh Rav of Rav Nassan Adler that Davened Nussach Arizal in his congregation of Nusach Ashkenaz, brought in Chasam Sofer O.C. 16; Nezer Hakodesh 2/12; Sefer Haminhagim p. 5 [English] footnote 16; Igros Kodesh 10/208; 14/391 [Published in Shulchan Menachem 1/171-174] However Tzaruch Iyun from a later dated letter in Igros Kodesh 19/4 in which the Rebbe directs a person who was asked this question that they are not to lead the congregation. Likewise, Rav Shusterman quotes a directive he received from the Rebbe in 1951 to Daven the Nussach of the congregation. Likewise Tzaruch Iyun from Igros Kodesh of the Rebbe Rayatz who gives a a public directive to Daven like the congregation. See Hiskashrus 694

Other opinions: Some Poskim rule that in all cases the Chazan is to Daven the same Nussach as the congregation. [Maharam Brisk 2/28; Meishiv Davar 17; Shaareiy Tzedek 8; Shoel Umeishiv Tilesa 1/247; Pardes Mordechai; Mishmeres Shalom 13/2; Igros Moshe 2/23; Minchas Yitzchak 7/5; Poskim in Piskeiy Teshuvos ibid footnote 29-30; Poskim in Igros Kodesh ibid] the reason or this is because of Lo Sisgodedu, or because the Chazan is the Shliach of the congregation.

[3] Igros Kodesh ibid; Nezer Hakodesh ibid

[4] Igros Kodesh Rebbe Rayatz 13/497; Regarding the practical directive of the Rebbe: a) The Rebbe in two letters and in Sefer Haminhagim permits and even encourages one to Daven his own Nussach if it will go without Machlokes. Thus, seemingly, if doing so will not lead to dispute, and certainly if he received permission from the congregation, he may Daven his own Nussach. If however it may lead to strife, then he is to follow the above public directive. See Hiskashrus 694

[5] Meishiv Davar ibid; Shoel Umeishiv ibid; Igros Kodesh Rebbe Rayatz ibid; Poskim in Nitei Gavriel 54/4 footnote 7

Other opinions: Some Poskim rule that even the private Shemoneh Esrei is to be prayed like the Nussach of the congregation, as itr is a preparation for Chazaras Hashatz. [Igros Moshe ibid; See Igros Kodesh 10/208; Piskeiy Teshuvos ibid footnote 30]