The Wiener Filter for the Time Derivative of the Signal
The key point here is the derivation / integration is a Linear Operator.
Since the Wiener Filter is also a linear operator it makes things easier.
Write the problem using the Wiener Filter Model:

Neglecting the integrator we know the Wiener Filter for $ \hat{H} \left( w \right) $ (See Wikipedia - Wiener Deconvolution):
$$ G(f) = \frac{1}{\hat{H}(f)} \left[ \frac{ 1 }{ 1 + 1/(|\hat{H}(f)|^2 \mathrm{SNR}(f))} \right] $$
By defining $ H \left( w \right) = \frac{1}{j w} \hat{H} \left( w \right) $ as the fourier operation of integration (This is not exactly accurate, see The Integration Property of the Fourier Transform)
$$ G(f) = \frac{1}{{H}(f)} \left[ \frac{ 1 }{ 1 + 1/(|H(f)|^2 \mathrm{SNR}(f))} \right] $$
Basically, we defined the integration as a filter and used the known building blocks.
The interpretation is, with respect to the non integral case, is apply the derivative operator on the estimator and have some regularization on the SNR estimation as we estimate a derivative.
Resources

How Is the Formula for the Wiener Deconvolution Derived?
Question on Wiener Filtering.
1D Deconvolution with Gaussian Kernel (MATLAB).