Let $C^*(\langle a,b \rangle)$ be the full group $C^*$-algebra of free group on two letters $\langle a, b\rangle$. Is there an algebra morphism $C(S^1)\to C^*\left(\langle a,b \rangle\right)$ which extends $id_{S^1}\mapsto a$ ?
Note that it is clear what to do on polynomials, namely $f(z)=\sum_k t_kz^k$ is mapped to $\sum_k t_k a^k$.
Your idea works in general. In a C$^*$-algebra you have continuous functional calculus on normal operators (and $a$ is a unitary), so you can define $\gamma:C(S^1)\to C^*(\langle a,b\rangle)$ by $$ \gamma(f)=f(a). $$ This works because, with $a$ a unitary, you have $\sigma(a)\subset S^1$ (it is actually an equality, but we don't need it here).