Never heard of this feud before… the idea of it sounds like a bit of a joke. Rebbe Nachman and the Alter Rebbe of Chabad were close friends and supporters of each other. When Napoleon invaded Russia all the Chassidishe rebbes came out in support of him while Rebbe Nachman and the Alter Rebbe were the only two Rebbes to side with the Litvaks on the side of the czar. This came in handy after the invasion was unsuccessful and the czar wanted to exile all chassidim – they were able to show that some chassidim had stayed politically loyal to him.
The Alter Rebbe once told Rebbe Nachman, “You are the Baal Shem Tov’s physical great grandson – I’m his spiritual one” As another poster mentioned the Alter Rebbe defended Rebbe Nachman against his detractors and conversely the Tanya was among the sefarim that Rebbe Nachman encouraged his chassidim to learn.
Fun side note – in Ramot there is a chabad minyan on shabbat that is at least 30-50% Breslovers and there’s a Chabad minyan with their own huge tent in Uman on Rosh Hashana.