Your skepticism is warranted. It's very difficult to prove a negative, but there is little to point towards this particular story of being true. All claims of this happening are either presented with no evidence or source beyond either Charles Henderson's 1986 Marine Sniper: 93 Confirmed Kills or Roy F. Chandler's 1999 White Feather: Carlos Hathcock USMC Scout Sniper -- an authorized biographical memoir. There is no information about this assignment from archival sources, for example, or other sources confirming that this actually occurred. While we could speculate and say that contemporary Vietnamese sources would try and hide such a humiliating mission which resulted in a general being killed, it is incredibly surprising that no official sources ever reference this mission or that any original material related to the mission has never resurfaced (while we simultaneously have plenty of available archival sources related to classified missions and units, including the controversial Phoenix Program). The only man that Hathcock, according to the two narratives above, told about this mission was his friend and fellow Marine Sniper John R. Burke who died in 1967 and is therefore not around to tell us what he heard from Hathcock.

Conveniently, all identifying details that could have made it possible to cross-reference this claim is missing, like you point out: What was the general's name? Where did this happen? When did this happen? We can assume that it took place before Burke death in 1967, since he's present in the narrative, but beyond that, we know nothing. In both Handerson's and Chandler's narrative about the mission, a great many cinematic occurrences take place that frames it as an invented scene from a war film rather than an actual military operation, including passages in which Hathcock can not get a clear shot because an aide or officer is in the way. Henderson is particularly egregious in his novel-like passages with exact dialogue from participants long since deceased. Once it was successfully carried out, why did it not become a propaganda triumph that could have been used by 'Free World' forces in order to humiliate and cast fear into their enemies while simultaneously raising morale within their own ranks? There are many questions but no ways of answering them.

It is therefore most likely that it never happened. While there is no doubt about Hathcock's abilities as a sniper, stories like these are very common amongst the mythification of snipers and marksmen in modern conflicts. Just consider all the stories attached to Finnish Second World War sniper Simo Häyhä. /u/Georgy_K_Zhukov has written this excellent post on the 'Cult of the Sniper' which covers much of this mythification. A few years back, /u/Lich-Su wrote this post on another claim that the question's OP and Hathcock made in regards to bounties.