Incredibly, the exact same thing (triangle piece fell out of my CHS) just happened to me 15 minutes ago!!!!!!!!
I was cleaning the watch & accidentally dropped it 2 inches onto the table, which caused the triangle to pop out & land between the dial & second hand, stopping the watch.
All your watchmaker needs to do is remove the caseback, remove the 2 screws/tabs that secure the movement into the case, remove the crown/stem & then remove the movement/dial assembly from the case. I used GC Crystal Cement, which every watchmaker should have on his bench, to glue the triangle back in place. If your local watchsmith is hesitant to perform this simple task, I would take your business to another watchmaker.