If McCoy was supposed to come across as awkward, it's partly that the entire dinner and the mission itself were supposed to be awkward in the first place.
It wasn't just that the dinner was awkward like it was supposed to be, it was awkward in the way it was edited.
So first of all why, other than to introduce the main antagonist, would Gorkon bring Chang along in the first place when it's clear to even the most casual observer that he hates the idea of peace and is probably going to act like a nasty little prick at this function. Does Gorkon have no judge of character at all?
So the dinner starts cordially enough, they toast Gorkon and to peace, make light of the fact they have Romulan Ale on ship and Gorkon toasts to "The Undiscovered Country", Spock catches the reference and then Chang starts his Shakespeare rantings. I know some people think Chang and his Shakespeare was the best thing since sliced bread, but to me it was just self indulgent tripe by Meyer to let the audience know how well read he is. But whatever.
I don't remember the exact order but Chang chides Kirk about giving up starfleet and tells him to be honest and Spock interrupts saying how Kirk believes starfleet's mission has always been one of peace and Kirk responds "Far be it from me to dispute my first oficer". Which I thought was pretty clever because it shows Kirk is really unhappy about this mission and is especially pissed at Spock for having put him in this position.
Then Chang repeats "To be or not to be" and says something about the Klingons needing breathing room to which Kirk says "Earth, Hitler 1938".......
To me this makes no sense and is just another Meyer "Look at how well I know my history" moment. First the whole point of this was not to expand the Klingon Empire so they had more "breathing room" it was to save their asses from almost becoming an extinct species.
Also if Meyer was trying to show how smart he was with this reference, he failed miserably. The concept of Lebensraum, or "living space" not "breathing room" far predated Hitler. It was first referred to in the late 1800's, used often as a reason in WWI and Hitler himself mentioned it countless times in his career, even in Mein Kampf, which came out in 1925. It wasn't like there was ONE time, like JFK saying "Ask not what your country can do for you......" where Hitler said it in 1938 in some iconic speech. It was a concept tossed around in Germany like "Independence" is in the US.
So then Chekov chimes in with his "We believe all species have an inalienable right..." comment. Which was pretty misplaced. It's not like he was totally off topic, but it was like if a group of people are talking about the Super Bowl being played tomorrow and then one person suddenly says "Yeah but the Packers need to get some help with the pass rush for next season".....Well it is about football, but doesn't fit the aspect that was being discussed. It just seemed so random.
Then Azetbur gets all wound up calling the Federation a homo sapiens club only....Uh aren't you the daughter of the man who is risking everything to pull this off, shouldn't you be more supportive of pops instead of insulting the people who are trying to save you?
Then Kerla puts in his extinction of the Klingon race which seems to get Bones wound up. Now if any starfleet character is going to lay it on the line and tell the Klingons they're acting like a bunch of total assholes considering THEY are the ones who came asking the Federation and help to save their race and the Federation is agreeing to help them, it would be Bones, the most passionate and hotheaded character in TOS. Spock sure is hell isn't going to do it, Kirk wants to do it but realizes as host and captain he can't, Chekov is a puss, Uhura maybe could have making some preachy connection between this and slavery, Scotty was half in the bag.....So Bones is the logical choice to stand up for the Federation and say "Now wait a damn minute." Instead he just mutely says "No"....and then Gorkon observes how far they have to go.
Also didn't Gorkon have ANY control over his people? Bringing Chang was asking for trouble. But couldn't he have said to Azetbur and Kerla "This dinner is important, we need this to happen, so put aside your feelings for the next two hours and keep your mouths shut and play nice?" If he did, they sure as hell didn't listen.
It was just a poorly edited scene that made the Klingons seem resentful that the Federation had the balls to actually help save the Klingon Empire. I understand awkward, but generally you don't want to belittle and insult the very people who are trying to help you.