Shotgun triggers are not like rifle triggers. I have hunted game birds and shot clay birds with shotguns that have heavy trigger pull...maybe in the 10-12 lb. range...and been very successful. I don't know if you were taught to squeeze a shotgun's trigger like a rifle, but I was not. I was trained to slap the trigger when the shot looked right, and I don't have a problem with missing. It's not a flinch, it's not a jerk, it is a straight slap to rear.
MY BSS trigger does not switch to the second barrel until it is released. Now maybe it's just a characteristic of those BSS guns that have been set up by a gunsmith for cowboy action shooting, because my fellow competitors have found the same characteristic. During a match if I hold the trigger back until after recoil settles, the trigger resets more slowly, or sometimes not at all. That costs time when shooting on the clock. So all the folks I shoot with who use a BSS in matches "pop" the trigger quickly, allowing recoil to assist the trigger return spring in switching to the second barrel. The triggers trip at about 3-4 lbs. pull and have a very smooth crisp let-off, but there is noticeable travel.
At this point I cannot explain it better than that. And I'm not interested in arguing about it. The technique works well for me and my fellow competitors in a cowboy action match. You can shoot yours any way you want.