Is there a simpler expression for $\sum_{j=0}^\infty 2^{I(j\neq0)} \cos j\theta \Gamma(2j+1) {\mathbb I}_j(\kappa) $, for large $\kappa$?

Question

We know that, when $\mathbb I_j(\kappa)$ is the modified Bessel function of the first kind of the $j$th order, and $I(\cdot)$ is the indicator function, then we have the relationship:

\begin{equation} \sum_{j=0}^\infty 2^{I(j\neq0)} \cos j\theta {\mathbb I}_j(\kappa)=\exp(\kappa\cos\theta) \end{equation}

However, I have a more complication with a Gamma term in there, and so the series:

\begin{equation} \sum_{j=0}^\infty 2^{I(j\neq0)} \cos j\theta \Gamma(2j+1) {\mathbb I}_j(\kappa). \end{equation}

Is there some simplication, or special function expression that I can get from here. I am especially interested in large values of $\kappa$.

Thanks in advance for any and all insights and/or suggestions!