Power: sufficient for reliable killing with a chest hit at 30 feet
Accuracy: .5 inch at 30 feet
Noise: as quiet as possible
Seems like a springer would do the job. They're usually the quietest, all else equal. The only iffy part is accuracy. Most of them will easily beat your accuracy goal from a consistent shooting position/hold. But when you are taking impromptu field shots, things can get squirrelly. I have a Crosman Venom, and despite a terrible trigger, it probably has the power and accuracy you would need.
The ole pump up Crosman 7xx rifle is great for most indoor pests, but chest shots on pigeons might be pushing the limit, even at close range. Depends how big they are. But from sub 30 feet, you could be taking head shots with some practice and a scope. Pumps and PCPs are dead easy to put the holes where you want them.
If you go PCP, you're looking at a lot of money and equipment, esp if you want it to be quiet. I think a Benjamin Marauder would be about the cheapest way to go, there.
Just cuz you are on a gun forum, I would also mention the possibility of using Aguila Super Calibris in a 22 rifle, depending on your local firearm laws. A 22 Savage rifle is around $100.00 at the Wmart. The Super Calibris are fairly accurate out to 30 feet, and about the only noise it makes is the sound of the firing pin. About the only air rifle that is cheaper is a pump up Daisy or Crosman. Power would be in the PCP rifle range, it's more quiet than any air rifle, and accuracy is probably on par with a springer in the field. Personally, I would feel more comfortable with my field accuracy at these ranges using the Super Colibris, over my springer, even.
Aguila Super Colibris are not to be confused with Aguila Colibris or Aguila Subsonics.