Improving Main Lobe Width of FFT
Given the phase coherence over non-blanked intervals, a simple and effective approach would be to additionally window the non-blank intervals with a matching window corresponding to the length of the time samples in between blanked intervals. Since the window only tapers the amplitude and does not modify the phase, there will be no issue with different window durations. This will significantly decrease if not eliminate the grating while provide phase coherent processing gain in the DFT from interval to interval over all usable samples. The Kaiser window would be a good window choice for this application. Consider how the DFT for any given frequency bin, as a correlation, de-rotates a signal that may be at that particular frequency such that each sample is aligned in phase, and then the result is summed; our best strategy would be to maximize the information from the samples that are there alone, rather than make our own attempts as to what we think is there to fill in the blanks, to then further convince our assumptions (unless we had other additional external knowledge, in which case such an approach could make sense).
The windowing approach would result with multiple various sized windows over the complete data set corresponding to intervals of non-zero data, and then an FFT would be performed on the entire data set.
To optimize this process further, Sparse FFT algorithms could be employed to take advantage of the gaps in time, but the windowing approach to modify the non-zero time domain data first would be functionally similar.