Probabilities of a Poisson distribution not making sense
This seems quite right to me. A poisson(1000) law has a standard deviation of $\sqrt{1000}$ which is something around $31.6$.

At such values, the Poisson law pretty much behaves like a normal distribution, so the probability of getting a value greater than 1100 would be very close to 0 (around 0.1%).

Perhaps what you need is not a Poisson law, but a truncated normal distribution?