Take any positive integers $n$ and $k$. Then
\begin{equation*}
Ee^{hS_n}=E\exp\{hW_{\min(n,\tau_k)}\}=s_1+s_2, \tag{1}\label{1}
\end{equation*}
where
\begin{equation*}
s_1:=Ee^{hS_n}1(\tau_k\le n)=e^{hk}P(\tau_k\le n), \tag{2}\label{2}
\end{equation*}
\begin{equation*}
s_2:=Ee^{hW_n}1(\tau_k>n)
=\sum_{j=-n}^{k-1}e^{hj}P(\tau_k>n,W_n=j). \tag{3}\label{3}
\end{equation*}
By the reflection principle, for each integer $j\le k$
\begin{equation*}
P(\tau_k\le n,W_n=j)=P(W_n=2k-j) \tag{4}\label{4}
\end{equation*}
(see details on this below), so that
\begin{equation*}
P(\tau_k>n,W_n=j)=P(W_n=j)-P(W_n=2k-j) \tag{5}\label{5}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
P(\tau_k\le n)&=P(\tau_k\le n,W_n\le k)+P(\tau_k\le n,W_n>k) \\
&=\sum_{j\le k}P(\tau_k\le n,W_n=j)+P(W_n>k) \\
&=\sum_{j\le k}P(W_n=2k-j)+P(W_n>k) \\
&=P(W_n\ge k)+P(W_n>k).
\end{aligned}
\tag{6}\label{6}
\end{equation*}
By \eqref{1}, \eqref{2}, the definition of $s_2$ in \eqref{3}, \eqref{6}, the latter equality in \eqref{3}, and \eqref{5},
\begin{equation*}
\begin{aligned}
Ee^{hS_n}&=e^{hk}P(\tau_k\le n)+Ee^{hW_n}1(\tau_k>n) \\
&=e^{hk}(P(W_n\ge k)+P(W_n>k)) \\
&+\sum_{j=-n}^{k-1}e^{hj}(P(W_n=j)-P(W_n=2k-j)).
\end{aligned}
\tag{7}\label{7}
\end{equation*}
The latter expression can be re-expressed in terms of the c.d.f.'s of the binomial distributions with parameters $n,\dfrac12$ and with parameters $n,\dfrac{e^{2h}}{e^{2h}+1}$ (see details below), and hence in terms of the incomplete beta function (see e.g. this comment).
Details on \eqref{4}: For each integer $j\le k$
\begin{equation*}
\begin{aligned}
&P(\tau_k\le n,W_n=j) \\
&=\sum_{t=1}^n P(\tau_k=t,W_n=j) \\
&=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n=j) \\
&=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n-W_t=j-k) \\
&=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k)P(W_n-W_t=j-k) \\
&=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k)P(W_n-W_t=k-j) \\
&=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n-W_t=k-j) \\
&=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n=2k-j) \\
&=\sum_{t=1}^n P(\tau_k=t,W_n=2k-j) \\
&=P(\tau_k\le n,W_n=2k-j) \\
&=P(W_n=2k-j),
\end{aligned}
\end{equation*}
since $2k-j\ge k$. $\quad\Box$
Details on re-expressing the latter expression in \eqref{7} in terms of the c.d.f.'s of the binomial distributions with parameters $n,\dfrac12$ and with parameters $n,\dfrac{e^{2h}}{e^{2h}+1}$: Note that $W_n=2B_n-n$, where $B_n$ a random variable with the binomial distribution with parameters $n,1/2$. Let $F_{n,p}$ denote the c.d.f. of the binomial distribution with parameters $n,p$. Then
\begin{equation*}
P(W_n\ge k)=P(W_n>k-1)=P(B_n>(n+k-1)/2)=1-F_{n,1/2}((n+k-1)/2)
\end{equation*}
and hence/similarly
\begin{equation*}
P(W_n>k)=1-F_{n,1/2}((n+k)/2).
\end{equation*}
Also, for $p_h:=\dfrac{e^{2h}}{e^{2h}+1}$ and
any real $a$ and $b$ such that $a\le b$,
\begin{equation*}
\begin{aligned}
&\sum_{a\le j<b}e^{hj}P(W_n=j) \\
&=\sum_{(n-b)/2<i\le(n-a)/2}e^{h(2i-n)}P(B_n=i) \\
&=\sum_{(n-b)/2<i\le(n-a)/2}e^{h(2i-n)}\binom ni 2^{-n} \\
&=\cosh^n h\,\sum_{(n-b)/2<i\le(n-a)/2}\binom ni p_h^i(1-p_h)^{n-i} \\
&=\cosh^n h\,\big(F_{n,p_h}((n-a)/2)-F_{n,p_h}((n-b)/2)\big).
\end{aligned}
\end{equation*}
So,
\begin{equation*}
\begin{aligned}
\sum_{j=-n}^{k-1}e^{hj}P(W_n=j)
=\cosh^n h\,\big(1-F_{n,p_h}((n-k)/2)\big)
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
&\sum_{j=-n}^{k-1}e^{hj}P(W_n=2k-j) \\
&=\sum_{i=k+1}^{2k+n}e^{h(2k-i)}P(W_n=i) \\
&=\sum_{i=k+1}^n e^{h(2k-i)}P(W_n=i) \\
&=\sum_{i=k+1}^n e^{h(2k-i)}P(W_n=-i) \\
&=e^{2kh}\sum_{j=-n}^{-k-1} e^{hj}P(W_n=j) \\
&=e^{2kh}\cosh^n h\,\big(1-F_{n,p_h}((n+k)/2)\big).
\end{aligned}
\end{equation*}
So, in view of \eqref{7}, indeed we can express $Ee^{hS_n}$ in terms of the c.d.f.'s of the binomial distributions with parameters $n,\dfrac12$ and with parameters $n,p_h=\dfrac{e^{2h}}{e^{2h}+1}$.
$\quad\Box$
