As a working definition I will define:
$$ K_\nu^{(n)} (z) = \frac{1}{2^n}\int_{\mathcal{P}} e^{- \operatorname{tr}( y + y^{-1}) z/2} \det y^\nu d \mu(y) $$
where $\mathcal{P}$ represents the space of positive definite symmetric matrices of size $n \times n$, $z \in \mathcal{P}$, and $d \mu_n(y) = \det y^{-(n+1)/2} dy$. When $n = 1$ they are the standard modified Bessel functions of hte second kind, and when $n \geq 2$ they are the matrix argument Bessel functions considered in C. Herz, "Bessel functions of matrix argument" (Zbl 0066.32002). When $n = 1$ and $\nu \in \mathbb{Z} + \frac12$ there are straightforward formulas giving $K_\nu^{(1)}$ as a sum of $e^{-z}$ and powers of $z$, and in particular,
$$ K_{1/2}^{(1)}(z) = \sqrt{\frac{\pi}{2 z}} e^{-z} $$
In Herz, it is shown that when $\nu \in \mathbb{Z} + \frac12$, $K_{\nu}^{(2)}(z)$ is expressible (essentially) as a sum of some $K_{\delta}^{(0)}(\operatorname{tr} z)$ with $\delta$ various integers, in particular:
$$ K_{1/2}^{(2)}(z) = \pi \det z^{-1/2} K_0^{(1)}(\operatorname{tr} z) $$
Herz accomplishes this by paramaterizing $\mathcal{P}_2$ as $$ y = \begin{pmatrix} e^\theta \cosh \psi & e^{(\theta + \varphi)/2} \sinh \psi \\ e^{(\theta + \varphi)/2} \sinh \psi & e^\varphi \cosh \psi \end{pmatrix}$$ and uses this to tear apart the defining integral for $K_\nu^{(2)}$ into an integral of a product of two $K_\nu^{(1)}$, and exploiting the product formula (see for example formula 10.32.18 in the DLMF).
My question is:
Are there similar explicit formulas relating $K_\nu^{(n)}$ to other functions when $\nu \in \mathbb{Z} + \frac12$?
Edit. As an added note, we have $K_\nu(z[k]) = K_\nu(z)$ for all $k \in O(n)$, so that $K_\nu$ depends only on $\sigma_1 = \operatorname{tr} z, \ldots, \sigma_n = \det z$, the elementary symmetric functions in the eigenvalues of $z$.
