Well, one of the possible reasons for pains in just those two fingers of the hand is an issue with the C7 nerve root as it comes out of the neck.
When a nerve is pinched in the neck, it can cause specific areas of pain.
C5 = shoulder
C6 = bicep and thumb/index finger
C7 = tricep and middle ring finger
C8 = elbow and pinky finger
So, when a person has complaints of pains in two adjoining fingers which seems unusual, it could be related to irritation of a nerve way upstream -- in the neck.
Carpal tunnel would have affected the thumb and index finger as well, so I would agree that is unlikely.
So, further testing could be helpful here, including being seen by an orthopedic surgeon or neurologist, and perhaps a MRI of the cervical spine if warranted.
Certainly the orthopedic surgeon can also rule out other issues specific with the hand (some of which have already been ruled out -- inflammation, gout, Lyme's disease, arthritis).