  Our primary goal in this article is to study the Iwasawa theory for
semi-ordinary families of automorphic forms on
$\mathrm{GL}_2\times\mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$, where $K$ is an
imaginary quadratic field where the prime $p$ is inert. We prove divisibility
results towards Iwasawa main conjectures in this context, utilizing the
optimized signed factorization procedure for Perrin-Riou functionals and
Beilinson--Flach elements for a family of Rankin--Selberg products of
$p$-ordinary forms with a fixed $p$-non-ordinary modular form. The optimality
enables an effective control on the $\mu$-invariants of Selmer groups and
$p$-adic $L$-functions as the modular forms vary in families, which is crucial
for our patching argument to establish one divisibility in an Iwasawa main
conjecture in three variables.
