Is the following distribution
$$P(\textbf{x}; \boldsymbol{\mu}, \boldsymbol{\Sigma}) = \int_{\text{h} \in \mathbb{R}^d} N\left(\textbf{h}; \boldsymbol{\mu}, \boldsymbol{\Sigma}\right) \,\text{Mult}\left(\textbf{x}; \pi(\textbf{h})\right) \text{d}\textbf{h}$$
where $\pi(\textbf{h}) = \frac{\left(1, e^{h_1}, \dots, e^{h_d}\right)}{1+e^{h_1}+\dots+e^{h_d}}$, $N\left(\textbf{h}; \boldsymbol{\mu}, \boldsymbol{\Sigma}\right)$ the multivariate normal distribution and $\text{Mult}\left(\textbf{x}; \pi\right)$ is the multinomial distribution, identifiable?
I.e. $$P(\textbf{x}; \boldsymbol{\mu}_1, \boldsymbol{\Sigma}_1) = P(\textbf{x}; \boldsymbol{\mu}_2, \boldsymbol{\Sigma}_2) \implies \boldsymbol{\mu_1} = \boldsymbol{\mu_2} \text{ and } \boldsymbol{\Sigma_1} = \boldsymbol{\Sigma_2}$$
I've found different sources dealing with a similar problem but I fail to find a final answer.
Lindsay, B. G. (1995). Mixture Models: Theory, Geometry and Applications. Institute of Mathematical Statistics (Vol. 5).
Chandra, S. (1977). On the mixtures of probability distributions. Scandinavian Journal of Statistics, 4, 105–112.
Hasselblad, V. (1969). Estimation of Finite Mixtures of Distributions from the Exponential Family. Journal of the American Statistical Association, 64(328), 1459–1471.
To know the values of $P(\mathbf x;\mathbf \mu,\mathbf \Sigma)$ for all $\mathbf x$ is to know all the mixed moments of the random vector $\pi(\mathbf h)$ and hence (because the range of $\pi$ is bounded) the distribution of the random vector $\pi(\mathbf h)$.
This follows from the algebraic form of the multinomial probability function: $P(\mathbf x|\mathbf \pi)= m(\mathbf x) \mathbf \pi^{\mathbf x}$ where $m(\mathbf x)$ denotes the multinomial coefficient for count vector $\mathbf x$ and ${\mathbf \pi}^{\mathbf x} =\prod \pi_i^{x_i}$. So the compound multinomial probability function $P(\mathbf x)=E[ P(\mathbf x|\mathbf \pi)] $ that you get when you take the expectation over $\mathbf \pi$ is thus equal to $m(\mathbf x)$ times the $\mathbf x$-th moment of the random vector $\mathbf \pi$. (The $\mathbf x$-th moment with respect to the mixing distribution for $\mathbf \pi$.) Finally, its a technical fact that the moments of a random vector distributed over a compact set, like the $d+1$ probability simplex, as your $\pi(\mathbf h)$ is, uniquely specifies the the distribution of that random vector.
So the question reduces to asking if the distribution of $\pi(\mathbf h)$ is identifiable.
Which it is: the map $\mathbf h\mapsto\pi(\mathbf h)$ is invertible, so to know the distribution of $\pi(\mathbf h)$ is to know the distribution of $\mathbf h$.
Why is $\pi(\mathbf h)$ invertible? We can recover $\mathbf x=(x_0,x_1,\ldots,x_d)$ by $h_i = \log(\pi_i(\mathbf h)/\pi_0(\mathbf h))$, when we number the subscripts $0$ through $d$.