The flu shot cannot make you sicker.
Here's the scoop - pharmaceutical companies try to anticipate which flu strains will be present during the flu season... and they linclude these strains in the flu shot.
Often times, like bacteria, viruses can mutate. The flu is a virus and thus has the potential every year to mutate. If it mutates, and that mutation of the strain is not inlcuded in the flu shot then the flu shot won't work.
If your child encounters a flu strain that is not in the shot they received, they'll get sick as if they didn't have the shot. Having the shot doesn't make anyone sicker.
People who say "oh I got the flu shot once and I got so sick that year. I haven't had it since and I never get sick" are using false causality. The flu shot they got that one year must not have covered the strain they encountered. And the years they didn't get the shot, they simply didn't encounter the flu virus.
The science is very straight-forward on flu shots.
PneumoVax (vaccine for Strep. Pneumo.) is to vaccinate against a type of bacteria. Although pseudo is the most common bacteria associated with CF, Strep. Pneumo is the most common cause of infection in "normal" people, including sinusitis, bronchitis, and pneumonia.
So let's say you, as a parent, get bronchitis one winter. Chances are its caused by a virus, or perhaps a bacteria. The most likely bacteria it will be caused by is Strep. Pneumo. Your kid will be protected against catching that bacteria from you if they receive a PnemoVax.
Hope that clarifies. Let me know if you have any more questions.
Bottomline is that your kid should get both.