I am not sure about the facts you mention, and I don't think I'll quite answer your question, but here are some facts I do know.

First, it is not the case that all $KU$-operations can be written as (even infinite) sums of Adams operations; Adams operations are additive, and general $KU$-operations need not be. So I won't say how to write the Chern classes in terms of Adams operations, but I'll try to say something about how they relate; I apologize if I am just repeating facts you are already familiar with.

One technical remark is that $KU$-operations aren't governed by $KU(BU)$, but really by $KU(\mathbb{Z}\times BU)$. I'll talk about $ku({-}) = [{-},BU]$ instead, which is valued in nonunital rings and has operations governed by $ku(BU) = \mathbb{Z}[[c_1,c_2,\ldots]]^+$, the augmentation ideal of $KU(BU)$. From now on, all my rings will be nonunital, but you can feel free to add a unit and think of them as augmented rings instead. I think my $c_i$'s are the same as yours; they are so that if we write $ku(BU(1)) = \mathbb{Z}[[u]]^+$ with $u=l-1$ with $l$ the tautological line bundle, then pullback under summation $ ku(BU)\rightarrow ku(BU(1)^{\times n})$ sends $c_i$ to the $i$'th elementary symmetric polynomial in $u_1,\ldots,u_n$.

Adams operations can be defined in any $\lambda$-ring, and $ku({-})$ is valued in $\lambda$-rings in the usual way. So you can write $\psi^n$ as a polynomial in $\lambda^1,\ldots,\lambda^n$, but not conversely. I think the relation is
$$
t\frac{d}{dt}\log(1+\sum_{n\geq 1}\lambda^n(x) t^n) = \sum_{n\geq 1}(-1)^{n+1}\psi^n(x)t^n,
$$
but I would not swear on the signs. Moreover, you can identify sums of Adams operations as exactly the additive operations acting on all $\lambda$-rings. Since this relates $\lambda$-operations and Adams operations, I'll just say how $\lambda$-operations relate to Chern classes.

For $\lambda$-rings like $ku({-})$ that are comprised of things like degree zero virtual bundles, it's useful to introduce the $\gamma$-operations. If I set $\lambda_t(x) = 1+\sum_{n\geq 1}\lambda^n(x)t^n$ for a formal variable $t$, and similarly define $\gamma_t$, then these are uniquely determined by asking first that $\gamma_t(x+y) = \gamma_t(x)\gamma_t(y)$, and second that if $\lambda_t(l)=1+lt$ then $\gamma_t(l-1)=1+(l-1)t$. You can explicitly relate these with the $\lambda$-operations via $\gamma_t = \lambda_{t/(1-t)}$, and a $\lambda$-ring structure is equivalent to a $\gamma$-ring structure.

Finally, a splitting principle argument lets you show that the class $c_n\in ku(BU)$ corresponds to the operation $\gamma^n$.