John upon reading your post, I am a little confused as to the gun going to safe with left barrel fired but not the right barrel. Do you mean that you cannot put the safe on manually or when the gun is broken open and re-cocked?

If you mean opening the gun and not having the safety automatically go to safe, it never will as there is no provision for that to happen. (see picture)
There is not enough room for the spur lever to rise and so there is no safety rod there.
If you mean that you cannot manually push the safety to safe without opening gun then there is a good chance that something is in the way on that side stopping it.


If you look at the safety in the picture, there is a lever that goes up to the safety and that relaxes after the trigger is pulled and it rests on that block that is sticking out next to it. I think something is stopping the lever from coming down to let the safety go to safe manually.
This is hard to diagnose without having gun in hand.