Consider random integer partitions $\lambda$ obeying the Poissonized Plancherel Measure of parameter $t^2$. We establish, through Riemann–Hilbert techniques, asymptotics of the multiplicative averages \begin{equation} Q(t,s)=\mathbb{E}\left[ \prod_{i\ge 1} \left(1+q^{s+i-\lambda_i}\right)^{-1} \right], \end{equation} for any $q \in (0,1)$ fixed and $s,t$ large. We compute explicitly the rate function $\mathcal{F}_q(x) = - \lim_{t \to \infty} t^{-2} \log Q(t,xt)$ which is expressed in a closed form through Weierstrass elliptic functions. The equilibrium measure of such a Riemann–Hilbert problem presents, in general, saturated regions and it undergoes two third-order phase transitions of different nature, which we describe. Applications of our results include an explicit characterization of tail probabilities of the height function of the $q$-deformed polynuclear growth and of the edge of the positive temperature discrete Bessel point process.