That's assuming the Klingons make an issue of it. Klingons have a bizarre culture, and they respect people defeating others, even defeating their own, in physical combat. They'll probably bizarrely declare M'Benga a powerful warrior who got revenge for his people in the Klingon way and leave it at that.
But he wasn't a Klingon national, he defected. He was as much of a Federation citizen as anyone else on that ship, the Klingons wouldn't care what happened to him no matter what, it's an entirely internal matter.
Though, I have to say, if I was an Alidar Jarok-type thinking these Federation people might be all right, the story of how a Klingon general defected, became the poster boy for how the Federation turns enemies into friends, and was a killed in a meaningless, seemingly unprovoked scuffle with a high-ranking doctor on the Federation flagship, I'd probably have some second thoughts about how reliable all that UFP propaganda really was.
This isn't some half-drunk rando stabbing Aamin Marritza on the Promenade for being a Cardassian, this was deliberate and knowingly done against a celebrity who the killer had met. I mean, this is nuts. This is like if a World War II vet stabbed von Braun in the '70s while they were both at work at NASA. It's wild, and Pike really should've been a bit more horrified rather than offering M'Benga a shoulder to cry on.
Also, as much as I complain about DSC and SNW not being able to remember that Klingons aren't are friends yet, having a Klingon defector be a coward, fraud, and genocidal maniac who needed killing by the ship's kindly old doctor is
way too far in the opposite direction.