Convergence of MGF of squared norm of sum of iid unit vectors

Question

Suppose I have $N$ iid random vectors $\sigma_1,\ldots,\sigma_N$ that are uniformly distributed in $S^1$. Let $\bar{\sigma}_N:=N^{-1}(\sigma_1+\cdots+\sigma_N)$ denote the sample average. I am interested in the asymptotic distribution of $X_N:=\left|\sqrt{N}\bar{\sigma}_N\right|^2$. More precisely, I am interested in a quantitative rate of convergence as $N\rightarrow\infty$ for the moment generating function $$\mathbb{E}[e^{tX_N}].$$ By the CLT, I know this (qualitatively) converges to the moment generating function $E[e^{\frac{t}{2}Q}]$, where $Q\sim \chi^2(2)$. Therefore, I only expect convergence to hold for $t<1$. Unfortunately, I am at a loss for how to get an explicit rate of convergence, i.e. an asymptotic expansion with respect to $N$ of $\mathbb{E}[e^{tX_N}]$.