I am looking for the tail bound of the following random variable, hopefully of sub-exponential form:

$\lambda_n^{-1}X_n\cos(\theta_n)$, where $X\sim Poisson(\lambda_n)$ with $\lambda_n\to 0$, and another independent random variable $\theta_n\sim Unif[0,2\pi]$.

So basically I'd like to know how much can that cosine factor "pull" the whole distribution back to its center, therefore improve the concentration result--since $\lambda_n^{-1}X_n$ as sub-exponential$(2\lambda_n^{-1},\lambda_n^{-1})$ random variable is poorly concentrated.

But this is so hard to compute: say I want to compute the MGF. If I first integrate over $\theta_n$, I get the modified Bessel function of the first kind: $$\mathbb{E}e^{\beta\lambda^{-1}X\cos(\theta)}=\mathbb{E}I_0(\frac{\beta}{\lambda}X),$$

which I don't know how to evaluate. Otherwise I get $$\mathbb{E}e^{...}=\mathbb{E}\exp(\lambda(e^{\beta\lambda^{-1}\cos(\theta)}-1)),$$

which is also bizarre.