Orthogonally additive holomorphic functions of bounded type over $C(K)$

It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form $$P(x)=\int_K x^k d\mu .$$ Thus $x\mapsto x^k$ factors all orthogonally additive polynomials through some linear form $\mu$. We show that no such linearization is possible without homogeneity. However, we also show that every orthogonally additive holomorphic functions of bounded type $f$ over $C(K)$ is of the form $$f(x)=\int_K h(x) d\mu$$ for some $\mu$ and holomorphic $h\colon C(K) \to L^1(\mu)$ of bounded type.