In that case, I assume a workaround (although computationally very expensive) would be to first perform a source inversion of the EEG signal, and then compute the regressions at source level?
You could try that, maybe using a few ROIs (not all the thousands of source time series).
But you probably would not obtain better results that the EEG directly. Maybe you'd get better results by using the two information in parallel, e.g. using your regression results to identify when you have elements of interest in the brain response, and visualizing the estimate sources at this latency.
What about arithmetic differences in EEG amplitude between 2 conditions? These are not 'actual' EEG signals anymore and might have very different statistical properties than the original data. is it also problematic to compute sources for such condition differences?
The subtraction is a linear operation, that can be permuted with the linear Minimum Norm source estimation: MNE(A)-MNE(B) = MNE(A-B).
This is explained for the average in the tutorials:
https://neuroimage.usc.edu/brainstorm/Tutorials/SourceEstimation#Averaging_in_source_space