Your pellet pistol will likely not have the power to kill pigeons, but could be used to finish a wounded walker. A .177( 4.5mm) Air Rifle with an advertised velocity of 1200 to 1500 FPS( these speeds are obtained with light alloy pellets that will damaged your springer or piston rifle. Do NOT use them.)with a heavy lead pellet the velocity will drop to between 900 and 1000 FPS. If your shooting outside of buildings you could use a .22 or .25 Calibre, but inside they will punch holes in things that should not have holes.
For example a 1250 rate rifle with the standard 7.6 grain pellet will be around 900 to 1000. If you have a high velocity you use pellets up to 10.6 grains. Which now have the energy to put holes in things that should not have holes.
Always have a backstop that will stop the pellet on a miss or pass-thru. Yes you can get a pass-thru with a .177 rifle. Straight through the chest hit both lungs and the heart, but only ribs and a thin lay of muscle on both sides, no breast plate or spine.