Good discussion; to which I would add these thoughts:
I would argue that "nudging" during benching with any appreciable weight would have limited practical effect for the tremendous risk. Not the least of which because the ability to "stabilize" when the spine is presumably pressed to the bench and therefore unable to move would completely isolate the shoulders. Further, the maximum stress point (the most difficult part) of this effort would be when the weight was most off center (and the shoulders are most vulnerable). No good could come from that.
Also, absolutely agree with BD19. If you are using stability balls, make sure they are tested for burst resuistance. Seen 'em blow up; it's just smart.
Finally, one of the best shoulder stabilization exercises I can suggest probably looks the easiest, but it is, arguably, one of the most difficult. Get two well inflated stability balls (easier on your wrists), put your feet on a bench, and do pushups with your hands on each. Great chest and shoulder workout!