This is a hard problem (and one which is easily overlooked by the unwary)! Just to be clear, I'll summarize (how I think about) the problem: as a relation between *additive* and *multiplicative* Fourier transforms. We start with a measure $\mu$, i.e. a linear map 

$\mu$: (continuous functions on $\mathbf{Z}_p$) $\to \mathbf{Z}_p$. 

For any $t$ with $|t| < 1$, there is a unique character $\kappa_t$ of $\mathbf{Z}_p$ as an *additive* group which sends 1 to $1 + t$; and we have $\mu(\kappa_T) = F_\mu(T)$ for some power series $F_\mu(T) \in \mathbf{Z}_p[[T]]$: the additive Fourier transform of $\mu$.

On the other hand, for each $u$ with $|u| < 1$, there is also a character $\chi_t$ of $\mathbf{Z}_p^\times$ as a *multiplicative* group, which is trivial on roots of unity and sends $1 + p$ to $1 + u$. This is given by $\mu(\chi_u) = G_\mu(u)$ for some power series $G_\mu \in \mathbf{Z}_p[[U]]$: the multiplicative Fourier transform of $\mu$. (I am deliberately using a different name for the variable here!)

Your question is, then, how $F_\mu$ and $G_\mu$ are related. The answer is: "not in any straightforward way". For a given $t$, the functions $\chi_u$ and $\kappa_u$ are totally different as functions on $\mathbf{Z}_p$, so the values of $F_\mu$ and $G_\mu$ at a specific $u$ do not determine each other. On the other hand, if $\mu$ is supported in $1 + p\mathbf{Z}_p$ (so we lose nothing by restricting), then the two series $F_\mu$ and $G_\mu$ do determine each other: we get a bijection (the "Mellin transform")
$$\mathfrak{M}: (1 + T) \cdot \mathbf{Z}_p[[(1+T)^p - 1]] \longleftrightarrow \mathbf{Z}_p[[U]]$$
which is a bijection of additive groups mapping $F_\mu(T)$ to $G_\mu(U)$ (but not respecting multiplication on either side). As an example, if $\mu$ is the Dirac measure sending a function to its value at $a$, for some $a \in 1 + p\mathbf{Z}_p$, then $F_\mu(T) = (1 + T)^a$, and $G_\mu(U) = (1 + U)^{log_p(a) / log_p(1 + p)}$. (This formula actually determines $\mathfrak{M}$ uniquely, since the Dirac measures are dense.)

One can pull various information about the power series through this isomorphism -- for instance, the Iwasawa $\lambda$ and $\mu$-invariants of the two series are the same, and the Newton polygons coincide beyond a certain point (Sarah Zerbes and I proved this in [an old paper of ours][1]). But there is no simple formula that would allow you to read off the coefficients of $F_\mu$ from those of $G_\mu$ or vice versa.

  [1]: https://dx.doi.org/10.1515/crelle.2012.012