How do I know for sure if this is buccal fat, muscle or something else? (Photo)

Bulge around the corners of the mouth is generally secondary to the junction of the muscles that animate the mouth ( modulus) and also a bit of excess of skin and sometimes a bit of excess of subcutaneous fat ( fat under the skin, NO residual buccal fat removal). Depending on the physical findings (causes) a treatment could be done. In case of saggy skin, skin tightening or cheek bone enhancement could improve the area. 

Buccal fat is an encapsulated pad located in the midline of the cheeks. Removal of buccal fat pads improves the contour of the face, and overall, doesn't age an individual. The positive of buccal fat removal is that the fat doesn't come back. I'm surprised to hear that your surgeon didn't get all of it the first time. The bulging near the cheeks is not likely residual buccal fat, it is most likely dispersed jowl fat or bulking muscle. Some of my patients have tried small amounts of Kybella in this region for fat reduction, though it is a notorious treatment for fat reduction under the chin. Before making a decision for any further treatments or costly procedure, I highly recommend consulting with a facial plastics surgeon because they have the adequate knowledge of facial anatomy and prospective treatments that will help you meet your goals.