I hate dealing with mouse corpses too. My go to for the house is a live trap and when one is caught I release it in with the chickens. I have never seen chickens fight so hard for something as when I release the mice.
For my shop I used a five gallon pail live trap. It involves a screened top with a 6" long narrow oiled tube in the center and bait in the bottom. The mice drop through the tube into the pail. They can jump out of an open top pail but the screen prevents that and the oiled tube (veg oil) is impossible to climb and too narrow for an accurate jump.
I don't have the pail trap now but here is where I first saw it.
https://youtu.be/ChpMSrBbn3I
In the fall on the farm I was catching up to 7 mice a night, even with farm cats. The chickens were very happy. I definitely see them as little dinosaurs when they catch and kill mice.
Edit: I used chicken feed or bird seed as bait so I could just dump all the contents in with the chickens.