The context in which I most often see the word disposition used, even in fairly casual contexts, is when discussing children. For example, in
You can't predict the weather when you are out there hiking, but you can predict your kid's disposition if the weather gets bad. (Advertising copy for a baby backpack rain cover)
Those first signs of your new baby's disposition are usually right! It isn't just your imagination—she really does have preferences from the first moment. (Holly Engel-Smothers & Susan M. Heim, Boosting Your Baby's Brain Power, 2008)
And conversation between parents:
There are always lots of kiddos being carried or on the side crying, so it depends on your DD's disposition, the day itself, your willingness to deal with it or carry her, etc. (Discussion board response to "Kids and Races: What Age?", Sept. 11, 2014; "DD" usually stands for "dear daughter" in these discussions)
Perhaps not coincidentally, it also shows up in descriptions of pets. For example, from the American Kennel Club:
The Airedale has a sweet disposition, but when challenged, is not afraid to stand up for himself.
And from the UK's Kennel Club:
Siberian Husky Breed Standards
Characteristics: Medium size, moderate bone, well balanced proportions, ease and freedom of movement, and good disposition.
As a result of seeing it used this way so often it no longer feels particularly formal to me. However, I still might hesitate just a bit in using it to describe adult humans, for fear that I might bring to mind babies or dog breed descriptions. Of course, folks who don't often discuss or read about small children or dog breeds probably wouldn't have those associations.