  It has been conjectured that for every claw-free graph $G$ the choice number
of $G$ is equal to its chromatic number. We focus on the special case of this
conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via
clique-cutset into two special classes called elementary graphs and peculiar
graphs. Based on this decomposition we prove that the conjecture holds true for
every claw-free perfect graph with maximum clique size at most $4$.
