Define $\mathcal S$ as the set of all $2^m$ skew-symmetric $2m \times 2m$ matrices with the form $\oplus_{j=1}^m\begin{pmatrix} 0&\pm 1 \\ \mp 1&0\end{pmatrix}$.

Let $S_i, S_j \in \mathcal{S}$, and $R^\prime, R^{\prime \prime}\in SO(2m)$. I'm looking for a function $f:SO(2m)\times SO(2m) \mapsto \mathbb{R}$ such that

$$ \int_{R\in SO(2m)} d\mu(R)\; \text{Pf}({R^\prime}^T.S_i.{R^\prime}+{R}^T.S_j.{R})\; f(R, R^{\prime \prime}) \propto \delta(R^\prime - R^{\prime \prime})\; \delta_{ij},$$ where $\mu$ is the uniform measure in $SO(2m)$, and $\text{Pf}$ is the Pfaffian.

Would you know of such a function, or have an idea how to go about constructing it? References? An existence (dis)proof would already be very welcome.

This would be a sufficient way to solve my previous question.