Not sure if it's your grip. Could be lack of upper back extension. After you inhale it looks like you are letting your chest sink. Also not sure if you are looking down too low. All these things tend to force you to support the bar with your arms instead of your back.
I had some minor tendonitis a while back and found that adjusting my grip didn't do anything, but focusing on my upper back helped. Especially the cue to pinch the shoulder blades together, like the photo in the bench press chapter of the blue book.