First, that's cool with the Benelli PG stock.
Second it is possible to double feed on 1201FP. When the bolt goes forward loading the round, the lifter will drop. If the little black button (at the end of the lift, in front of the trigger guard) does not catch the lift, then next round will load on to the lift.
Try this with dummy rounds, put one in chamber, and one in tube. Press the little black button, ah next shell loads. Try again, but this time put two dummy round in the tube. Pull the trigger (shell load) slowly pull back the bolt, eject the shell from the chamber, now hold the black button back and let the bolt go. Yeap, you'll have a fresh round on the lift as the bolt closes.
If the black button is worn, it is possible for double feed. It is also possible to have junk stuck behind the button, causing it not extend fully. Take the trigger group out and clean can help.
With that, I still cannot see how the next shell will affect the bolt close completely. You may have other issues.