Can the following be of interest?

We know that

\prod_{n=1}^N \mathsf{Poisson}(y_n \mid \lambda_n) \ = \ \mathsf{Poisson}(\mathrm{sum}(y) \mid \mathrm{sum}(\lambda)) \cdot \mathsf{Multinomial}( y \mid \lambda / \mathrm{sum}(\lambda)),

but that’s just a different way of stating the same distribution on y.