Do you REALLY think the President, Congress, the military and, hell, the American people are just going to shrug and say, "Whadda you gonna do?! Send them a truck full of food every couple of weeks and we'll call it good."
Well that's what happened during Katrina (and Sandy), so yes I do buy it.
As I recall, the American people were VERY upset with what happened with Katrina. (And the fairly minor missteps with Sandy are hardly comparable to the week-long clusterfuck of Katrina. Unless the media has just not mentioned rapes and murders happening in a New Jersey stadium.)
So what?! People get upset about a lot of things but what you fail to realize is that people's attention span is pretty short. People would get angry for a week or two and then the news cycle moves on.
Also slow response to a natural disaster in a city is very different than a large region being held hostage by a nuclear weapon over a period of MONTHS.
You asked if government could be totally apathetic. I gave you a real world example.
I'm sorry, but Bane's takeover of Gotham is just not comparable to the Hurricane Katrina response. It's completely fair to say that the federal and state governments responded lethargically to Katrina, but that was a function, frankly, of class and racial biases -- the majority of people left in New Orleans were poor black folk, after all; the wealthier white folk to whom the governing elite tend to feel the closest kinship had escaped. And while the government's response was lethargic, it was not nonexistent.
Meanwhile, Bane's takeover of Gotham resembles, quite frankly, nothing that has ever happened in modern history. You are, after all, taking about the violent overthrow of a municipal government and the taking of millions of people hostage with a nuclear bomb, by a foreign terrorist organization.
The closest comparison we can realistically draw upon would be 9/11 -- a single day's attack on three buildings in two cities. Not an entire city taken hostage for five months. 9/11 happened 11 years ago, and the country
still hasn't gotten over it.
So the idea that "[p]eople would get angry for a week or two and then the news cycle moves on" is just absurd. People don't just forget about their most important city being cut off from the world by a foreign terrorist because Kim Kardashian is pregnant. This is the sort of thing that would profoundly change national politics and culture, that would traumatize the country for decades afterwards.
But, that doesn't mean that the federal government's response to Bane's actions in
The Dark Knight Rises is itself implausible. After all -- Bane cut off all their options very quickly; anything they could possibly do openly would almost inevitably end in the destruction of the city and the deaths of millions. And their main attempt at covert action ended in failure (because of Miranda/Talia's presence among the corporate hostages).
Personally, I think it's completely plausible that the federal government would essentially try to appease a Bane-like figure in an attempt to avoid the instant nuclear death of millions. (Though, of course, being that no terrorist organization has ever taken a city hostage with nuclear weapons before, we can't know for certain.)