From Adams, we know that the algebra of (unstable, degree-zero) cohomology operations $K^0(BU)$ can be written as formal infinite linear combinations of canonical generators

$$\mu_n := \sum_{i=0}^{n} (-1)^{n-k}\binom{n}{k} \psi^k$$

However, from the collapse of the Atiyah-Hirzebruch spectral sequence for BU, we also know that $K^0(BU) \cong \mathbb{Z}[[c_1^K,c_2^K,\ldots ]]$ where $c_i^K$ are the Conner-Floyd Chern classes (where I'm renormalizing them to degree zero by an appropriate power of $t\in \pi_2KU$).

Thus, it should be possible to write the Chern classes in terms of the Adams operations. How can I find these expressions?

Doing the reverse is not so bad, using Hirzebruch's theory of genera: I get that $\psi^k$ is $(1+c_1^K+c_2^K+\ldots)^k$. But unfortunately I'm completing lacking in the power series wizardry that would allow me to invert this.

This was wrong, since the multiplication of characteristic series of genera does not correspond to anything over on the cohomology operations side, coinciding neither with cup product nor composition (which are themselves distinct).