I’ve read that mech triggers have an inertia block to prevent “doubling”
https://www.trapshooters.com/threads/inertia-blocks-in-mechanical-triggers.806569/
Edited to add
I checked numrich and I couldn’t find a part called an inertia block
I'm not anywhere near an expert, but according to the responses in your linked thread, the Browning 725 trigger being used as an example is not a "true mechanical" trigger. "It clicks without recoil but does indeed require recoil force to reset the lifter to the second sear."
For your shotgun, the information I linked describes it as a selectable mechanical trigger. You can set either barrel to fire first, and then the other will fire with the second trigger pull. If the first selected barrel fails to fire, you can then select and fire the other one.
I can't find any schematics of the trigger parts for that model. It's discontinued, so who knows if that's online.
I'd say try switching the barrel selector a few times to see if the reportedly non-firing barrel will fire as the first one selected. If the selector makes no difference, it could be a problem with that sear, a spring, or even that firing pin. Tough to analyze if the only info you have is second hand.
If you have some 12g Snap cap rounds (I assume that's the right caliber), you could put tape over the primer area of a couple Snap cap shells, and see if you are getting a primer strike from each firing pin. Might be easier than waiting to see if it goes "BANG!" at the range.