Well, I wouldn’t compare it to Sophie’s choice, since that was a Satanic - and utterly specious - proposition that the Nazi officer put to her - he wasn’t going to spare one child, he simply wanted to degrade and humiliate her even more by making her “choose” which of her children was going to die.
In this situation, the disturbed child is so miserable that being institutionalized is not going to be significantly worse for him, but his absence from the home could mean the beginning of a normal life for the other two children.
Perhaps not the best comparison, but by institutionalizing her other son, she will probably rob him of a normal life. Not that he would have one anyway, but as long as he’s at home, she probably holds out hope.
I would probably go ahead and commit him, but such a decision would definitely take its toll.