CLT doesn't say anything about the number of elements in the sum. You can have a sum of 1000 variables from Poisson(0.001) and CLT won't say anything about the sum. All it does say is that if you keep increasing N then at some point this sum will start looking like a normal distribution $\frac{1}{N}\sum_{i=1}^N x_i, x_i\sim Poisson(0.001)$. In fact if N=1,000,000 you'll get the close approximation of normal distribution.

Your intuition is right. The more formal (but still informal) way would be by looking at the characteristic function of Poisson: $$\exp(\lambda (\exp(it)-1))$$
If you $\lambda>>1$, you get with the Taylor expansion (wrt $t$) of the nested exponent:
$$\approx\exp(i\lambda t-\lambda/2t^2)$$
This is the characteristic function of the normal distribution $\mathcal{N}(\lambda,\lambda^2)$