  We continue with our study of the non-critical exceptional zeros of Katz'
$p$-adic $L$-functions attached to a CM field $K$, following two threads. In
the first thread, we redefine our (group-ring-valued) $\mathcal{L}$-invariant
associated to each $\mathbb{Z}_{p}$-extension $K_{\Gamma}$ of $K$ in terms of
$p$-adic height pairings and interpolate them as $K_{\Gamma}$ varies to a
universal (multivariate) group-ring-valued $\mathcal{L}$-invariant. In the
second thread, we use our results to study the exceptional zeros of the
non-genuine Rankin--Selberg $p$-adic $L$-functions attached to the
self-products of nearly ordinary CM families, via the factorization statements
we establish. The factorization theorems are extensions of the results due to
Greenberg and Palvannan.
