In Europe most 626s are double triggers, so there should be parts for the conversion at some dealers or Beretta itself.

Seeing that the system is not resetting itself, yet it can be manually changed to the second barrel, it seems that something is blocking the inertia block from moving back under recoil. Check if you can freely move the inertia block back. A worn pin, or some debris trapped somewhere under the block might be the problem.

The inertia system is so simple that it cannot be anything major.