Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.
That $T$ has closed range $R(T)$ is equivalent to the existence of a constant $C > 0$ such that
$\forall y \in R(T) : \exists x \in T^{-1}(\{y\}) : C \| y \|_H \geq \|x\|$
Another equivalent condition is that the Moore-Penrose inverse of $T$, written $T^\dagger$, has a norm bounded by the constant $C$.
Do you know a norm bound for $(i^\ast T i)^\dagger$? You assume that $G$ is finite dimensional so that $i^\ast T i$ has closed range.
