Hi, flatland:
Duh! I should have looked for this parts diagram first.
http://www.gunuts.com/gunuts/images/16.gif
OK, the Inertia Block, #PO32272, resets the trigger so you can fire the 2nd barrel. It's on the upper left side of the diagram, below the selector safety button, #PO32300. It looks like it's right under the safety button on the assembled action. It's sticking or some part connected to it has a problem (broke or sticking). I can't tell how it works from the diagram, but the trigger has to trip the sears, #PO32300 and #PO32320, so some part has to shift from one sear to the other for the second shot.
Do a little dry-firing with the butt off. Pull the trigger to fire the first barrel, wiggle the inertia block back and forth, and try the second barrel. Hopefully you can see what's wrong.
Bye
Jack