There is a question of freedom to act. In the example, when a mother steals an apple to fit the child, I indeed did not clarify the presumption that for a mother there is no choice but to feed the child. If she has a choice: the child is crying but not in danger of death, then she may not grab that apple. Another example would be if the private property is not marked: then, perhaps (depending on the acculturation) an intruder is not acting freely simply because he does not know that the apple is private and cannot be taken. (The latter was the situation with the American Indians who simply could not grasp the concept of private land for agricultural use).
Back to your example: the firefighter is a professional, trained to evaluate the situations and consequences of them. So, he will be killing the fat woman freely by pushing her off the stairs. His duty is above all not to kill anyone; this duty remains even though she is about to die anyway. His duty to save the children is satisfied: he made an attempt and discovered that he could not do so without committing a murder. If the child pushed the woman, or a parent, then the moral equation would have been different because the child or parent would be acting in a highly agitated state and not freely.
Likewise, a doctor may attempt to save the life of the mother in labor, but he may not kill the baby in order to treat the mother: both the baby and the mother are his patients.
Likewise, a priest who is told at confession that the penitent is about to commit a heinous crime may not use the information the priest obtained to stop the crime.
These are all cases when double effect provides no excuse because a positive step is taken freely and in the knowledge that the step is immoral. The fact that the criminal step serves a greater good does not excuse the step.
I should have clarified, too: that under no circumstances can the firefighters kill the fat woman, as a means or as an end. Even if they can't save her, she has to be treated as a dying person, not a sack of meat.
For instance, if they found they couldn't shove her because of her weight, they couldn't dismember her and throw the pieces down separately. That would be murder full stop.
However, if the landing point is only one floor (10 feet) down, you can quickly make a reasonable judgment that hitting the floor won't kill her. You might (professionally) anticipate that it could cause broken bones, depending on how she hit, but that doesn't absolutely rule out an action: EMT's sometimes break ribs doing CPR on frail patients, people sometimes break legs landing in a rescue net, it's a commonly acknowledged risk, justified (I think) by the proportional and reasonable expectation that this will save lives.
As I mentioned also, it would be morally obligatory, if possible, to try to ensure a safer impact (throw a mattress down first) and, later, to rescue her, of course, again if it's possible.
But in my view, just pushing her off the steps in itself is not murder. There is not the intent of murder, and it is not an act of murder. It is an act of triage. (In my view. I am not an expert in the situations that arise in fire rescue situations, for sure.)
I would agree that just shoving her off as a "dead loss" when she's not dead, would be gravely wrong.