Do German hangs with your hands in whatever position they want to go to...that's the benefit of rings.
Internally rotating the shoulders creates impingement situations, are you purposely trying to internally rotate in the German hang? If so, I'd say stop doing that.
Bulgarian dips internally rotate the shoulder a lot. That obviously bothers you.
I would tend to be externally rotated in a German hang, or the more common "stretching" exercise equivalent of reaching behind to grab a bar at shoulder height (both arms behind the body) and squatting down to stretch the front of the shoulder/chest. This is typically done with palms facing the ground, and that's putting the shoulders in *external* rotation, opening the chest, freeing up the shoulder joint. Doing that same move with the palms up (like I think you're basically trying to do) puts the shoulders in internal rotation, closes the chest, and puts the shoulders in a great position for impingement.
I hope this makes sense. Let your shoulders go where they want to in the German hang is the take-home message, I guess...
I'm guessing that you allow the rings to turn in on ring dips, thus aggravating your internal rotation issue. The reason why p-bar dips likely don't aggravate your shoulder is because, besides the added stability, you are not "allowed" to internally rotate your shoulder as much due to the bars.