A Poisson distribution with a mean of 3158.6 is not necessarily wrong and may fit your data well, but it's difficult to work with from a practical point of view. For example, if you want the probability of exactly 3158 deaths, that would be given by
which is hard to compute. So although a Poisson distribution may be theoretically correct in this case, it's difficult to work with, and for that reason it would probably be better to use a normal distribution as an approximation.
By the way, it's not correct to say that e^-3158.6 is zero-- it's just very small (despite what your calculator may say).