I do believe Khârn would be bested in this particular case; and here is why
-The swarmlord has the toughness, the size and the physical and metaphysical protection to endure several cuts from gorechild without flinching. Although usually the drawbacks of such a tough creature are a lack of agility and speed, it is stated in the fluff that the swarmlord lacks neither; being able to parry with his sabres at lightning speed and all that. Because of this Khârn, while Khârn will have a harder time to land a telling blow and properly wound the beast even ignoring all the Swarmlord's armor, the creature's ability to block strikes (With 4 arms!! Against 1). And it's size and mass means that even if it suffers a hit, there is no guarantee it will kill it.
-The swarmlord has the offensive resources: As mentioned above he has 4 arms wielding equally lethal weapons with a strength far superior to that of even an augmented superhuman, the size, sharpness, and special characteristics of the blades only further enhance their lethality. The swarmlord can attack from all directions using his blades, which makes in incredibly difficult to avoid an attack, and it can even attack with a pair while defending with another, or some similar "offense-defense" tactic. We have to consider that a single blow from the swords would be more than capable of tearing Khârn apart or severely maiming him if he gets caught (and with the insane melee skill of the SL, it's not that far-fetched to suppose he can land a blow). One could argue that Khârn's frenzy, bloodlust, and unholy might would allow him to ignore the pain and fight on, but losing an arm or being thorn in half would make this will to go on irrelevant, as his melee capabilities are horrendously diminished
The fact that the SL's weapons are so lethal if only a single one of them hits home (and let's not forget there's 4 of them) would require Khârn to be aware of the position of the blades at all times, and ready to avoid or counter all of them. This is somewhat contradictory with his usual fighting style, which is more akin to a butchering frenzy without much regard for defense. And I'm not sure if he would be able to adapt to this new fighting style. Although it could be argued that he is just as capable of maintaining a cold head and a defensive stance if necessary; as he is depicted in the Heresy series as very calm and calculating even in the midst of battle. But I don't know how much of that cunning, tactical Khârn is left after 10000 years in the warp.
Even if Khârn is the single best human champion in the galaxy, he is constrained by the fact that this is no duel between equals, he is fighting a monstrous creature of equal cunning and skill, but much greater toughness and strength than him.
All in all, the need for Khârn to land several strikes to kill or maim the beast, while the SL only needs one or maybe two successful swings to severely limit Khârn's combat capability. As well as the fact that Khârn can only strike from one angle and one direction at a single point in time, while the SL has 4 attacking options at any given moment, give the Swarmlord an important edge in an even fight.
That is not to say Khârn would die; He would probably lose, but not finished. He has been retrieved from death several times by Khorne, and even if he was bested he could be retrieved by his warband or simply saved by Khorne's grace to join the Long War once more.
Sadly, any author would much rather kill the Swarmlord, as it would give COOLNESS FACTOR to Khârn, and it can simply be respawned by the hive mind and is thus expendable.