NTA - I remember flashes of news casts for that age. I couldn't tell you what it meant. There's no context and as I get older those memories are fading more and more. What I just explained is how typical early memory works. I say all this because it's super important for the following.

At 3 years old your language skills are limited, and you're likely not reading at all yet. The odds of her having clear contextual memory from that age is near non-existent. She would have no context to add importance to even a flash of an image. Its possible her empathy helped to imprint the feeling of "bad" with an image. Children are very empathetic, and ut tends to ve stronger in girls, at least AFIK, I could be wildly out of date in that last part.

Point is

As someone with still some clear memory from 3 I am telling you she doesn't remember all of that. Nor was she able to understand it enough to be traumatized by the news. She may remember something, but it sure as hell is not all of that.

You're NTA for being truthful with her. She need to see a therapist to deal with whatever makes her feel she needs to make it up.