Limited pain is letting you know something needs attention.
Have you done voodoo wrap?
Are you addressing the movement pattern(s)which bother it?
Have you contacted a health practitioner to help you put together an exercise plan to address this situation?
Movement is a great part of treatment.
Push up and bench are the press archetype.
Are you able to create the press archetype with no pain?
Internal rotation of the shoulder may be restricted.
Episode 89: Bro, Your Internal Rotation is Lacking
Episode 315: Great New Shoulder Fix-Internal Rotation Tack and Stretch
Daily Prog: Shoulder Internal Rotation 1/3
WOD: Shoulder Internal Rotation 2/3
WOD: Shoulder 3/3, Internal Rotation and Function