Every so often I'll start the car and notice that the S light on the
dash is on, but the car is not in sport mode! It's not blinking, so
it's not trying to throw an error.

As far as driving the car, when the S light is on the acceleration
becomes very slow, and the transmission will not downshift no matter
how fast the car is going. Pushing the sport button on the shifter does
nothing.

The problem goes away after a few minutes of driving, however. As in
the S light goes off and the car accelerates and shifts normally.

Could this be a computer issue or a symptom of a larger problem?