Orthogonality of $L^2(G)$ for a compact group

Question

I am having trouble with the following:

Let $G$ be a compact group with Haar measure $\mu$ and $(\rho,\mathbb{C}^n)$ be an irreducible representation of $G$. Naturally, we can suppose that $\rho$ is unitary by Weyl's trick. Fix a $\rho(G)$-invariant inner product $\langle-,-\rangle$ on $\mathbb{C}^n$.

Suppose $f\in L^2(G:\mathbb{C}^n)$ is of the form $f(g)=\rho(g)v$ for some $v\in \mathbb{C}^n$. Let $h\in L^2(G:\mathbb{C}^n)$. Does there exist $w\in \mathbb{C}^n$ such that $\int_G \langle f(g),h(g)\rangle d\mu=\int_G \langle f(g),\rho(g)w\rangle d\mu$?

I believe the answer should hold by Peter-Weyl, but I am struggling to do the relevant calculations. Could someone please verify?

Thanks!