...a healthy-looking mouse corpse in the flower bed behind my house...

I'd bet it was sick/died of old age/froze to death, and that you don't have anything on the property desperate enough to risk taking a bite-- it might have an odor of sickness repelling scavengers. Eventually the bugs will eat it, or it will turn to mush and be washed into the ground leaving a nice tiny skeleton you can display as a trophy of your mouse hunt! :~).   A little glass cheese-dome-display case might make an interesting knick-knack!

~S~