How do you "split apart" the output of a linear estimator?
There is nothing wrong about subtracting the bias. For the Weiner filter, the operation is a matrix multiply, there are no non-linear operations taking place which is reason everything works out. There are two possible ways to do it:

Subtract the bias before the filter. In this case you'll have a Weiner filter, $G_1$, which will produce $\hat{x}$ given $x+n$.

\begin{align}
\hat{x}&amp;=G_1(y-c)
\end{align}

Subtract the bias after the filter. In this case you'll have a Weiner filter, $G_2$, which will produce $\widehat{x+c}$ given $x+c+n$ (this is the case you presented).

\begin{align}
\widehat{x+c}&amp;=G_2(y+c)\\
\hat{x}&amp;=\widehat{x+c}-c
\end{align}
Note: Be careful with the notation of $G$. The Weiner filter is constructed using the data so the filter constructed using the $x+n$ data will be different from the filter constructed using the $x+c+n$ data.