Kill it as if I were going to feed it to my snake, then drop it in the woods well away from any houses. Lots of critters out there that would welcome the meal, and I'd rather not have a mouse rotting anywhere near my house for any length of time.
I don't think any animal should die and then serve no purpose but to rot quietly somewhere. At least some part of the animal, if not every part, should be used in some way, even if it's only as food for something else, or to be buried in a garden as fertilizer, or set on fire in honor of its life. To die because you were in the wrong place at the wrong time, and then to have that death be utterly meaningless and be left to rot in a trash heap where nothing can benefit from your sacrifice.... It's almost blasphemous.