Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this:

Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and let $\rho$ be an irreducible representation of a group $G$ over $V$. Taking $\epsilon \geq 0$, if for all $g \in G$ it is the case that $\| [ B, \rho(g) ] \| \leq \epsilon$, then there exists a constant $\lambda \in \mathbb{C}$ such that $\| B - \lambda I \| \leq f(\epsilon)$, where $f$ is some ''nice'' function of $\epsilon$ (obviously we want that $f(\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$).

Basically I'm asking: if $B$ almost commutes with the irrep does that imply that $B$ is close to a multiple of the identity (in operator norm)?