John, sorry to hear that you spent so much money and did not get results. There are only about 4 people in the United States that know anything about a Hunter One Trigger (HOT), the rest can tell you that they can work on your L.C. Smith but again there aren't many gunsmiths that know the workings of one.
There are only a few parts to a HOT that can break and most of the time they do not work is because someone took one apart and did not know how critical it is to put it back together. There are three (3) screws that can make a HOT not work, the most common one is the side plate connecting screw (the one that goes through both lockplates), one half a turn either way and this trigger will not work correctly. The threaded end should be flush with the right lockplate, the screw that is under the top lever, this screw is orientated to the gun and most times it is one complete turn too tight. What happens is that too tight causes the trigger plate to rise up and brings the sears too close to the trigger plate causing the gun to have a hair trigger or fire both barrels. The other screw that can cause problems but not as much is the screw under the trigger guard that goes up into the top tank behind the trigger. This screw should also be flush with the top tang.
When ever I take one of my guns with a HOT apart I always measure those three areas before disassembly and then again putting it back together.


You stated that when you open the gun between shots the safety works securely, do you mean that you pull the safety to Safe?
There is a piece of spring steel that has two detents in it under the safety and you can hear when the safety passes over it. This keeps the safety in place when on Safe and off. Possibly one leg is broken but the gunsmith should have noticed it.