Your upper bound on $\mathbf{E} \left[ \exp\left(\theta R\left(X+X'\right)\right)\right]$ is $$f(a,b;t):=(\cosh at+bt^2)^2+\sinh^2 at,$$ where $$a:=EX\ge0,\quad b:=Ee^X-e^{EX}\ge0,\quad t:=\theta.$$ You want to upper-bound $f(a,b;t)$ by $e^{st^2/2}$.

Expanding $f(a,b;t)$ and $e^{st^2/2}$ into powers of $t$ and comparing the coefficients of the same powers of $t$, we see that $$s=4(a^2+b)$$ will do.

Your upper bound on $\mathbf{E} \left[ \exp\left(\theta R\left(X+X'\right)\right)\right]$ is $$f(a,b;t):=(\cosh at+bt^2)^2+\sinh^2 at,$$ where $$a:=EX\ge0,\quad b:=Ee^X-e^{EX}\ge0,\quad t:=\theta.$$ You want to upper-bound $f(a,b;t)$ by $e^{st^2/2}$.

Expanding $f(a,b;t)$ and $e^{st^2/2}$ into powers of $t$ and comparing the coefficients of the same powers of $t$, we see that $$s=4(a^2+b) \tag{10}\label{10}$$ will do.

Indeed, we have $$f(a,b;t)=\sum_{k=0}^\infty c_k t^{2k},\quad e^{st^2/2}=\sum_{k=0}^\infty C_k t^{2k},$$ where $$c_0:=1,\quad c_1:=2(a^2+b),\quad c_2:=\frac{2 a^4}3 + a^2 b + b^2,$$ $$c_k:=\frac{a^{2 k-2} \left(a^2 2^{2 k}+4 b k (2 k-1)\right)}{(2 k)!}\text{ for }k\ge3,$$ $$C_k:=\frac{(s/2)^k}{k!} \text{ for }k\ge0.$$

It is straightforward to check that $c_0=C_0$, $c_1=C_1$, and $c_2\le C_2/2$ given \eqref{10}. It is also easy to see that $\dfrac{c_k}{C_k}$ is decreasing in $k\ge2$ given \eqref{10}. Thus, the claim follows. $\quad\Box$

Moreover, since $c_0=C_0$ and $c_1=C_1$ given \eqref{10}, we see that the expression for $s$ in \eqref{10} is the best possible one.