“All hope was lost when, unlooked for, the Riders came out of the North and broke upon the rear of the enemy. Then the fortunes of battle were reversed, and the enemy was driven with slaughter over Limlight. Eorl led his men in pursuit, and so great was the fear that went before the horsemen of the North that the invaders of the Wold were also thrown into panic, and the Riders hunted them over the plains of Calenardhon.’ The people of that region had become few since the Plague, and most of those that remained had been slaughtered by the savage Easterlings. Cirion, therefore, in reward for his aid, gave Calenardhon between Anduin and Isen to Eorl and his people; and they sent north for their wives and children and their goods and settled in that land. They named it anew the Mark of the Riders, and they called themselves the Eorlingas; but in Gondor their land was called Rohan, and its people the Rohirrim (that is, the Horse-lords).

LOTR Appendices - as will be every quote in this post

2644-2718TA [In the reign of] Déor. In his time the Dunlendings raided often over the Isen. In 2710 they occupied the deserted ring of Isengard, and could not be dislodged.

“There was at that time [the reign of King Helm, 2691 - 2759] a man named Freca, who claimed descent from King Fréawine [who had reigned 2594-2680], though he had, men said, much Dunlendish blood, and was dark-haired. He grew rich and powerful, having wide lands on either side of the Adorn.

There was at that time a man named Freca, who claimed descent from King Fréawine, though he had, men said, much Dunlendish blood, and was dark-haired. He grew rich and powerful, having wide lands on either side of the Adorn. Near its source he made himself a stronghold and paid little heed to the king. Helm mistrusted him, but called him to his councils; and he came when it pleased him.
‘To one of these councils Freca rode with many men, and he asked the hand of Helm’s daughter for his son Wulf.

‘Then Freca fell in a rage and reviled the king, and said this at the last: “Old kings that refuse a proffered staff may fall on their knees.” Helm answered: “Come! The marriage of your son is a trifle. Let Helm and Freca deal with it later. Meanwhile the king and his council have matters of moment to consider.”

‘When the council was over, Helm stood up and laid his great hand on Freca’s shoulder, saying: “The king does not permit brawls in his house, but men are freer outside”; and he forced Freca to walk before him out from Edoras into the field. To Freca’s men that came up he said: “Be off! We need no hearers. We are going to speak of a private matter alone. Go and talk to my men!” And they looked and saw that the king’s men and his friends far outnumbered them, and they drew back.
‘“Now, Dunlending,” said the king, “you have only Helm to deal with, alone and unarmed. But you have said much already, and it is my turn to speak. Freca, your folly has grown with your belly. You talk of a staff! If Helm dislikes a crooked staff that is thrust on him, he breaks it. So!” With that he smote Freca such a blow with his fist that he fell back stunned, and died soon after.
‘Helm then proclaimed Freca’s son and near kin the king’s enemies; and they fled, for at once Helm sent many men riding to the west marches.’”

‘To one of these councils Freca rode with many men, and he asked the hand of Helm’s daughter for his son Wulf. But Helm said: “You have grown big since you were last here; but it is mostly fat, I guess”; and men laughed at that, for Freca was wide in the belt. ‘Then Freca fell in a rage and reviled the king...

Where Helm errs, in my view, is that even if he didn't intend to murder Freca, the very public and violent nature of Helm's version of 'cutting down to size', followed up by declaring Freca's entire household to be "the king's enemies", ie. outlaws, suddenly purges the west marches of its ruling elite.

I've long read Helm and Freca coming to blows and the subsequent events as a proxy for the faultline that runs through the founding of the Kingdom of Rohan. The feigned historians involved, although not challenging the legitimacy of that foundation do sometimes acknowledge that the Dunlendings are bearing a centuries-old grudge stemming from their displacement from the western parts of what was then Gondor's Calenardhon province. What became known as the Westfold became the fulcrum of that grievance.

What the Helm vs Freca vignette does is reinforce / re-tell the ethnic tensions that run through this displacement / foundation story. As Eldy remarked, Helm goes for what is clearly an ethnic slur or trope, when insulting Freca as a 'Dunlending'. Where this episode differs is in the consequences for the Rohirrim: for the first time, the Eorlingas are defeated in force by the Dunlendings and their allies (at the Crossings of the Isen), and expelled from their capital, Edoras. A king, albeit an usurper, sits in Meduseld who the Dunlendings actually support - although it's worth noting that Wulf, or at least Freca, claimed descent and legitimacy from Rohan's fifth king, Fréawine.

As for the specifics, not only does Helm lean right into the ethnic trope, he behaves more like a thug than a king who's dynasty had reigned for nine generations and presumably could rely on more than just the violent, very personal settling of slights, I reckon. He literally wants to assert his authority through a one-punch brawl that pretty much guarantees another round of conflict - unless Wulf can be captured, executed or otherwise permanently exiled.

“Now, Dunlending,” said the king, “you have only Helm to deal with, alone and unarmed. But you have said much already, and it is my turn to speak. Freca, your folly has grown with your belly....

'You don't belong here; you're no Baggins - you - you're a Brandybuck!'

'Did you hear that, Merry? That was an insult, if you like,' said Frodo as he shut the door on her.

'It was a compliment,' said Merry Brandybuck, 'and so, of course, not true.'

- A Long Expected Party