You are probably looking for an instrument that plays in the lower range, but the specific choice may depend upon what the "grumbly old bear" is doing.
A bass or cello would be the obvious choices among the string family of instruments. By using various bowing techniques to add unusual qualities to the tones being produced, you could convey a bear stretching, scratching, and wiggling back to a comfortable position to continue hibernating.
A baritone saxophone, bass clarinet, or other low wind instrument would be another possibility. These instruments could allow for squeaks and other sounds that might be useful to emphasize sudden changes in direction if the bear was running or catching a fish.