JD Jones beat you to it Nolo.  .300 Whisper.

I guess you could abrogate the Hague Convention and go to the C. Anvil "XX Special" projectile, with high explosive charge AND dose of poison in case the blast and impact did not kill the opponent.

The BPCR crowd did some work with subsonics, holding the .45-70 down to 1000 fps.  They soon reverted to the traditional 1200fps because test trumped theory and they got better accuracy even with the Mach transition downrange.

JD Jones beat you to it Nolo.  .300 Whisper.

What does the terminal effectiveness of .300 Whisper (which I am pretty familiar with) look like? I've never seen a study on it? What mechanics does it use to do damage to tissue, etc.?
AFAIK, 9x39mm is designed specifically to penetrate armor. I've heard that it "drops Chechens like a rock", but that's only anecdotes of anecdotes, so it can't be relied upon.
I am not saying .300 Whisper is the right answer, I am just wondering what its terminal ballistics look like.

Nope.  It's not a problem.  Accuracy is fantastic.  And the down range impact - it hits harder at long range than 5.56mm does.  Higher BC helps a lot.
Nolo - if you want to learn about Ballistics - JDJ is the Yoda.
The best thing about the .300 Whisper is that you can run it in your AR with just a different upper.  You can use your same mags.

I don't know enough about fluid mechanics, but don't blunt-nosed teardrop shapes have the best BC when they're subsonic?

No.  BC is BC weather or not it's above or below the sound barrier.  Drag is Drag.  Blunt nosed bullets do have one advantage... which is why slippery pointy boat tails are not always the best choice.  Tissue disruption. That wider ogive when it hits - disrupts more tissue.  It's not a stabbing cut like a pointy bullet.  It's a blunt force trauma that penetrates. 
Look at the bow of a ship.  Say, a sleek sail boat.  It cuts the water with ease and glides through leaving a wake like an arrow-head.  That wake is the water being pushed away.  So the boat, with its sharp lines, loses less energy, so it needs less energy to move through the water.
Now look at the bow of an Ice Breaker or a Ferry.   Wide and blunt.  Look at the water piling up against the bow.  Look at the wakes.  These ships really have to push a lot of water out of the way.
Air behaves the same way.  Air is essentially, a thin fluid.  It moves the same.   Get a cigarette and blow some smoke.  And pass your hand through it flat and edged and look at how it behaves.   Go out to a lake and take a cannoe paddle.  Run it through the water the same way... flat and on edge and look at how the water moves.  The same. 
Now, as you know... tissue is like fluid too. 
Now with tissue you get other neat things.  Bone shards, tendons, blood, etc. 
The .300 Whisper doesn't depend on shock like the 556 does.  It uses good old fashioned penetration.  Momentum and large permanent wound channels.

Much appreciated, George. That was highly informative.
I noticed in a ballistic calculator that I used that the same bullet (.357 250-grain boattail, simulating 9x39mm) going subsonic (1040fps) loses 25% less energy at 500 (or was it 1000? I can't remember) yards over a supersonic (2000 fps) bullet.
This was surprising to me, but it seems to fit with your statement.
Again, thank you.

So now the question becomes, .300 Whisper? 9x39mm? .338 Spectre? Etc, etc.

Nolo - When you are looking at those balistic results... make sure that you factor in heavier bullet weights with the higher BC's.  Heavier means longer, and longer tends to mean higher BC.  But not always.  Such as my Beloved 7mm... which is almost always a higher BC than other calibers unless those other calibers go with very heavy bullets for caliber... ahem... anyways... Look at the Hornady 154 grain SST and compair that to the 162 grain BTSP.  The 154 has a higher BC, even if it is shorter... and the BTSP is pointy... but look the bullet over carefully.  Can you see where the drag is?
Now tell me.  How is the 154 SST slicker?  This is subtlety.  Look carefully.  The difference between the two in BC is slight.  But its there.
This might give it away...
Now look at the 162 A-Max' BC.
I do love that bullet.
But it's useless for hunting. 

Just for the convenience of those of us trying to see what George is talking about:

Hornady 154gr SST, BC .525:

Hornady 162gr BTSP BC .514:

The secant ogive is important.  Now loot at the positions of the cannelure.
The A-Max doesn't have one.

Disruption is just as important as the shape of the nose.

This is why the Air Force was playing with Bottle Shaped, or Wasp Shaped aircraft fusilages.  Not because of the stuff that they needed to fit under the skin - but how it moved through the air.   This is also why Howard Hughes had his speed planes with "Shaved" Rivets.  Less disruption, less drag.

Speed is Life.  Drag is too.   To retain that down range energy - you want as little drag as possible.  A higher BC.

I am assuming the bullet with the more forward cannelure has a higher BC because the cannelure gets caught in the shockwave and doesn't affect performance as much?

Hahah, my dad has a Doctorate in Aeronautics and Astronautics. I pick up things sometimes.

See, this page doesn't show any recovered bullets in subsonic ranges, suggesting that they don't perform well terminally under the sound barrier.

I like the idea of a silent rifle, but I don't want it to just put a .308, .338 or .356" hole in my enemy!