Let $Γ$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $Γ$ (with respect to some finite subset $F \subseteq Γ$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $Γ$ is the free group, answering a question of Meehan.