Contrasting @BESW's opinion somewhat, I'd say the answer to your core question, namely: "what other rules do I absolutely need to run a traditional CoC game?", is, simply:
You don't need any other rules besides the obligatory skill checks (have a skill of 0-99, roll d100, if the result is equal or smaller than your skill rating, you succeed, if not, you fail) ...if you're not very bad at rule-improvising, and your party is willing to accept your judgment.
I'm talking from hard experience here. We've been playing CoC one shots in a variety of half-official, half-homebrew settings (ranging from the 1890s to the 2390s) for almost a decade with such a super-minimalist, rules-be-damned approach, and I can count on the fingers of one hand the number of times we checked some rule (like piercing) in the book.
Of course, this means you'll have to improvise some rulings -- but the "how likely is that to happen (in percentages)? well, roll that now!" rule of thumb works 99% of the time for practically anything.
Let me repeat: I'm talking from years of real experience of running CoC one shots here. (And I'm not trying to boast here -- what would be the point in that? --, I hope that's obvious. I'm not even saying my solution beats any other, nor that everyone would like our stories. Who knows? Maybe they'd be considered plain bad by some. But we've been having fun for years. So, for some, a similar approach may work.) Practically a d100, and a good and fair gut sense of approximating the chances of something is all you need. :)
To add a few thoughts on the points the Q raises, at BESW's request:
One-shot game, highly dangerous for characters
Danger is inherent in most stereotypical CoC encounters. Besides good storytelling (which is the primary device of raising a sense of being endangered in any horror game), adding tough but just and appropriate (negative ;)) modifiers to a d100 skill roll is well enough. If you decide to keep the HP system, scaling lethality to your needs is a breeze: Personally, I'd recommend keeping the current HP (and, possibly, Sanity) of the players' characters a secret from the players, issuing warnings and subtle reminders about their condition and its changes as the story progresses.
"You raise your gun with a swing, trying to aim at your poor brother's head, but you just can't keep your arm up. Adrenaline rush or no, it's most likely broken. You might try and use your left, of course."
A few of the investigators probably dabble in occult arts
Use Sanity as designed, if possible. The Q itself says it's essential, so I'm assuming OP knows these rules. If not, though, you can go for a simple improvised d100 roll here as well. After all, on the grand scale, Sanity loss is the mental equivalent of suffering physical injury.
"Your head is still spinning from what you've seen on the terrace. But what was it? You don't want to remember that. Not now. Not ever. You want to stay inside for the rest of your life. Inside is safe. Outside is your brother who's become the shadows. All of them. Even though that's impossible. Even though it's your own doing. So, are you going to lock the doors?"
Very light on combat; investigators are encouraged to do whatever
they can to avoid straight confrontation. They only have a chance to
actually beat the enemy versus humanoids.
See above, under danger, mostly. Roll your skill to hit, if successful, tell and note the effect it has on the opponent. If possible, keep current HP and roll damage in secret. Tweak the damage values of weapons etc as you see fit.
As with combat, not much action going on, though I can see something like one really good and appropriate car chase
Again, use your sense of drama and the core mechanic, a d100 test, with appropriate modifiers. :)