Let $X$ be a nonnegative random variable such that $\mathbf{E} \left[ \exp X \right] < \infty$. For $\theta \leqslant 1$, an appropriate application of Jensen's inequality, yields that

\begin{align}
\mathbf{E} \left[ \exp \left( \theta X \right) \right] \leqslant \exp \left( \theta \mathbf{E} \left[ X \right] \right) + \theta^2 \left( \mathbf{E} \left[ \exp X \right] - \exp  \left( \mathbf{E} \left[ X \right] \right) \right).
\end{align}

As a consequence, I can obtain the following bound for the moment-generating function of $R \cdot \left( X + X^\prime \right)$, where $R$ is a standard Rademacher random variable, and $X, X^\prime$ are independent copies of $X$ (now for $| \theta | \leqslant 1$):

\begin{equation}
\mathbf{E} \left[ \exp\left(\theta R\left(X+X^{\prime}\right)\right)\right] \leqslant \left(\cosh\left(\theta\mathbf{E}\left[X\right]\right) + \theta^{2} \left(\mathbf{E}\left[\exp X\right] - \exp\left(\mathbf{E}\left[X\right]\right)\right)\right)^{2} \\
+ \sinh^{2}\left(\theta\mathbf{E}\left[X\right]\right).
\end{equation}

I would like a simple upper bound on this quantity, ideally of the form $\exp\left( \frac{s}{2} \theta^2 \right)$, and with a reasonably sharp constant $s$. One can check that this is the right behaviour around $\theta = 0$, and "large" $\theta$ are not really of interest anyways (since we only go as far as $\theta = \pm 1$ anyways), and so the resulting bound should be quite usable and probably even not too loose.

I have considered taking the logarithm of this quantity, differentiating this twice with respect to $\theta$, and then trying to upper-bound this uniformly, but this seems likely to get messy. A less painful solution would thus be desirable.