No
I could find no other instance of dwarves leaving a battle after their leader was killed. The closest I can get is at the Battle of Sarn Athrad, where Beren slew the Lord of Nogrod, but it is very unsatisfactory.
In fact, there are many counter examples of dwarves staying in the battle: Thorin dies in the Battle of Five Armies, but "the dwarves were making a stand still about their lords upon a low rounded hill"; Dáin died in the Battle of Dale, but the dwarves continued to fight. Also, the War between the Dwarves and the Orcs was kindled by the death of Thrór, the King of Durin's Folk.
The timing of this event is also important: it happens at the end of Nirnaeth Arnoediad, the Battle of Unnumbered Tears. At this point, the battle is basically lost. It may be that the death of Azaghâl, the original wearer of the Dragon-helm of Dor-lómin, was too much for them to bear. It could also be that this was the one battle where they could get away with it: Glaurung and his brood were all running away from them at this point:
[...] with his last stroke Azaghâl drove a knife into [Glaurung's] belly, and so wounded him that he fled the field, and the beasts of Angband in dismay followed after him.
The Silmarillion, Of the Fifth Battle, p.229
Battle of Sarn Athrad
After dwarves from Nogrod slayed Thingol over the Nauglamír, Beren leads an army of Green-elves and kills the Lord of Nogrod himself. The dwarves did flee, but it is unclear whether it was because their Lord was killed. In any case, there is no singing here, just a lot of running away. The timing is also unclear, so I'm not sure whether the death of the Lord of Nogrod happens during the battle or if it ends it.
There very many of the Dwarves were slain in the first onset; but some escaping from the ambush held together, and fled eastwards towards the mountains. And as they climbed the long slopes beneath Mount Dolmed there came forth the Shepherds of the Trees, and they drove the Dwarves into the shadowy woods of Ered Lindon: whence, it is said, came never one to climb the high passes that led to their homes.
In that battle by Sarn Athrad Beren fought his last fight, and himself slew the Lord of Nogrod, and wrested from him the Necklace of the Dwarves; but he dying laid his curse upon all the treasure.
The Silmarillion, Of the Ruin of Doriath, p. 282
War of the Dwarves and Orcs
Thrór, at the time poor and desperate, but still King of Durin's Folk, tried to reenter Moria alone, but was killed by Azog, an Orc.
When [Thráin II, son of Thrór] had wept and torn his beard he fell silent. Seven days he sat and said no word. Then he stood up and said: ‘This cannot be borne!’ That was the beginning of the War of the Dwarves and the Orcs, which was long and deadly, and fought for the most part in deep places beneath the earth.
The Lord of the Rings: The Return of the King, Appendix A: Annals of the Kings and Rulers, III: Durin's Folk, p. 437-438
The war culminates with the Battle of Azanulbizar and although the dwarves win the war, they do not rejoice:
But no feast nor song was there that night; for their dead were beyond the count of grief. Barely half of their number, it is said, could still stand or had hope of healing. [...] ‘We fought this war for vengeance, and vengeance we have taken. But it is not sweet. If this is victory, then our hands are too small to hold it.’
The Lord of the Rings: The Return of the King, Appendix A: Annals of the Kings and Rulers, III: Durin's Folk, p. 440
Death rituals
Dwarves do seem to be heavily affected by losses in battle and they have somewhat elaborate burial processes: They bury their dead in stone tombs only. After the Battle of Azanulbizar however, there were too many bodies to make this practical:
So it was that after Azanulbizar the Dwarves dispersed again. But first with great labour they stripped all their dead, so that Orcs should not come and win there a store of weapons and mail. It is said that every Dwarf that went from that battlefield was bowed under a heavy burden. Then they built many pyres and burned all the bodies of their kin. There was a great felling
of trees in the valley, which remained bare ever after, and the reek of the burning was seen in Lórien.
The Lord of the Rings: The Return of the King, Appendix A: Annals of the Kings and Rulers, III: Durin's Folk, p. 440
A note to this paragraph says this:
Such dealings with their dead seemed grievous to the Dwarves, for it was against their use; but to make such tombs as they were accustomed to build (since they will lay their dead only in stone not in earth) would have taken many years. To fire therefore they turned, rather than leave their kin to beast or bird or carrion-orc. But those who fell in Azanulbizar were honoured in memory, and to this day a Dwarf will say proudly of one of his sires: ‘he was a burned Dwarf’, and that is enough.
ibid.
