  Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$
over a totally real number field $F$. Let $K$ be a totally imaginary quadratic
extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times
\operatorname{GL}_2$ Rankin-Selberg $L$-functions associated to $\pi$ times
representations induced from Hecke characters of $K$ which are ramified only at
a given prime ideal $\mathfrak{p}$ of $F$. More specifically, we use spectral
decompositions of shifted convolution sums and relations to Fourier-Whittaker
coefficients of genuine and non-genuine metaplectic forms to obtain
nonvanishing estimates, averaging over primitive ring class characters of a
given exact order. When $\pi$ corresponds to a holomorphic Hilbert modular form
of arithmetic weight $k \geq 2$, we then derive finer results from the
rationality theorems of Shimura, together with the existence of suitable
$\mathfrak{p}$-adic $L$-functions. This allows us to generalize the theorems of
Rohrlich, Vatsal, and Cornut-Vatsal to this setting. Finally, in a
self-contained appendix, we explain how to use these results to deduce bounds
for Mordell-Weil ranks of the associated $\operatorname{GL}_2$-type abelian
varieties via existing Iwasawa main conjecture divisibilities.
