I bought a 20 ga. BSS in 1978, choked full and modified. Took it to a Browning approved gunsmith, who shot it with a 1 oz. lead load and determined it to shoot 73% full, 69% modified. Typical tight Browning chokes.
He cut out the modified barrel to 35%, and said it was some of the best steel he had ever cut in a gun. He said there should be no problem with steel shot. He warned about large steel shot that could bridge at the constriction of the full choke and push through the shot collar. So I have stayed away from anything larger than #4 steel.
I have handloaded 3" Hevi-Shot in #6 and #4 with Ballistic Products components in Fiocchi hulls. Works fine, deadly on tough wild Iowa pheasants. I have handloaded #2 Hevi-Shot for a friend which he uses in his 20 ga. A-5 Magnum with imp. cyl. Invector chokes. No problems, though I warn him to keep checking
whether chokes turn out hard that would indicate peening of the threads from the big shot. He had been using #1 Federal steel loads to shoot early season geese in the face without any problems. (His calling allows that).
My 2 cents worth: 1) As long as you stay with smaller shot, there should be no problem at all with steel shot in your BSS. 2) Shoot Hevi-Shot at wild pheasants (if you can stand the cost). It cuts down the number of runners. 3)Use the same chokes as you would for steel.