  Fix an integer $\kappa\geqslant 2$. Let $P$ be prime and let $k> \kappa$ be
an even integer. For $f$ a holomorphic cusp form of weight $k$ and full level
and $g$ a primitive holomorphic cusp form of weight $2 \kappa$ and level $P$,
we prove hybrid subconvexity bounds for $L \left(\tfrac{1}{2}, \text{Sym}^2 f
\otimes g\right)$ in the $k$ and $P$ aspects when $P^{\frac {13} {64} + \delta}
< k < P^{\frac 3 8 - \delta}$ for any $0 < \delta < \frac {11} {128}$. These
bounds are achieved through a first moment method (with amplification when
$P^{\frac {13} {64}} < k \leqslant P^{\frac 4 {13}}$).
