I hunt in an area where I need to keep the noise down too, so understand where you are coming from. I hunt Red Deer with a .300 Whisper and have never lost one or even come close to it. I use the Hornady 160gn FTX which I load to the max and get very close to 30/30 ballistics. It doesn't use a lot of powder so less powder less boom

I've experimented for years trying to hunt subsonic and it is just not, I believe ethical. There is not a bullet made that expands consistently for subsonic hunting so it's basically like hunting with solids. The 160gn FTX gives good expansion at supersonic speeds and all the energy is expended in the animal without any exits. (on the animals I've shot).
As a side note, I haven't taken a shot over 200m on game with it and I think that's its absolute max sensible range.
Failing that, there is nothing wrong with a .223 and it will tip over your wee deer effortlessly. I've shot quite a few Fallow in NZ with a .223 and they don't go anywhere when hit.