Let $W_N(β) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N βω(n,S_n) - Nβ^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $ω(i,x), i\in \mathbb{N}, x\in \mathbb{Z}^2$ are i.i.d. standard normal and $\{S_n\}$ is the path of a random walk. With $β=β_N=\widehatβ \sqrt{π/\log N}$ and $\widehatβ\in (0,1)$ (the subcritical window), $\log W_N(β_N)$ is known to converge in distribution to a Gaussian law of mean $-λ^2/2$ and variance $λ^2$, with $λ^2=\log ((1-\widehatβ^2)^{-1})$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb E [W_N( β_N)^q]$ in the subcritical window, and prove a lower bound that matches for $q=O(\sqrt{\log N})$ the upper bound derived by us in Cosco, Zeitouni, arXiv:2112.03767 [math.PR]. The analysis is based on appropriate decouplings and a Poisson convergence that uses the method of ''two moments suffice''.