IRL Henry Eaton died about a month after, so feelings had cooled down considerably and there was little tension with the Maori over it. If the men had found the warriors though, and the incident was more bloody than IRL, then there could have been more consequences.
The Kahungunu had been at war with the settlers allies the Te Ati Awa as recently as the year before, and had raided and killed Te Ati Awa chief Puakawa in February, so perhaps it could have ended with a resumption of that war, only this time with European involvement. If they invade the Kahungunu territory in the Wairarapa, and perhaps defeat them and claim the land, it would speed potential colonisation as IRL the land wasn't bought until the mid 1850s.
Also on the 14th, Captain Pearson of the ship Integrity was brought to the NZ Company's court over a dispute regarding some cattle he'd been chartered to bring over, IRL he didn't recognise the authority of the court, since the company was running things as if they were a republic, and not under British/Colonial authority, and ended up escaping, then sailing north and telling the Lieutenant-Governor about the Company's independent rule. This ended up with troops being sent down in June, to tear down the United Tribes flags and force the Company settlers to swear oaths of allegiance.
If there had been a battle that same day, maybe the guards would be more watchful, and Pearson would fail to escape- meaning it would at least take longer for the colonial government to realise what was going on in Britannia, which would allow the Company to have more time to get further entrenched and build more infrastructure. By the time time the troops arrived IRL they already had a Constitution, Council, President, Court system, Constabulary and beginnings of a Militia, with the town having a bank, multiple pubs, and a few other industries setting up. They didn't recognise the Lt Governors proclamation of sovereignty as applying to all of NZ, so it would be interesting to have them stay independent for longer.