This is only a partial answer, but perhaps it will help. While the symmetric functions $\Lambda$ are graded by polynomial degree, the shifted symmetric functions $\Lambda^*$ are only filtered. It is fairly easy to see from the definition (and proved in Okounkov-Olshanski's paper *Shifted Schur Functions*) that in fact the associated graded algebra of $\Lambda^*$ with respect to this filtration is isomorphic to $\Lambda$,

$$\text{gr}(\Lambda^*) \cong \Lambda.$$

In particular, in terms of polynomial degree the shifted Schur function $s_\lambda^*$ and shifted power sum function $p_{\lambda}^*$ can be written as

$s_\lambda^* = s_\lambda +$ l.o.t.,

$p_\lambda^* = p_\lambda +$ l.o.t.

So if we only look at the top degree terms of your $f_\lambda$ we get exactly $s_\lambda$, but in general $f_\lambda \neq s_\lambda^*$. I don't know if anything is known about the lower degree terms of $f_\lambda$.

I should mention that Okounkov-Olshanski introduce another shifted analogue to the power sum functions which they denote by $p^\#_\lambda$. These do have the property that,

$$s_\lambda^* = \frac{1}{n!}\sum_{\mu \vdash m} C_\mu\chi^{\lambda}(\mu) p^\#_\mu.$$

$p^\#_\mu$ also has the wonderful property (at the expense of having a nice explicit formula) that if $\mu \vdash k$ and we evaluate $p^\#_\mu$ on a partition $\lambda = (\lambda_1,\lambda_2, \dots, \lambda_r) \vdash n$ by setting $x_1 = \lambda_1, x_2 = \lambda_2, \dots, x_r = \lambda_r$, and all other $x_i = 0$, then we get

$$p_\mu^\#(\lambda) = \begin{cases}
\frac{(n \downharpoonright k)}{\dim L^\lambda} \chi^\lambda(\mu) & k \leq n,\\
0 & \text{otherwise}
\end{cases}$$

where $(n \downharpoonright k)$ is the falling factorial and $\dim L^\lambda$ is the dimension of the simple $S(n)$-representation associated to $\lambda$.