Maybe not so much a delt problem, as a rotor cuff problem. Let us hope not. Or quite often a pinched or stressed nerve can cause this. The nerves become inflamed from all this. Might suggest a deep sports massage, from time to time. Keep the shoulder warm while working out. Can apply a sports lotion/rub to the area before and after working out. With the idea to keep the area warm and with good blood flow. Try a stretch and hold before working out. As Chaos suggest, rotor cuff movements by all means. Can not hurt but only help. Even changing the hand position on the bar a few inches may give some relief when doing BB BP's.
Drop the BP for a while. Even DB fly's can influence delt pain. A Pec Deck may be a better tool for you, the the pressure on the elbow pads, not with the hands/arms way out. If you believe the myth that BP'ing is the only way to go for chest, than switch to DB benches. Advantage is that the hands/muscles/joints are not fixed in one position during the pressing. DB's allow a more natural twist and turning of the hands/wrist/elbows and shoulder girdle. Less stress all the way around.
Might consider dips as a better chest mass producing exercise than benching. Worth a try. Not sure if you meant that inclined did or did not bother you. If Ok than might do:
Dips, inclines and pec decks for the chest. Good Luck.
