Moiré No More
I would think the optimal technique lies somewhere between the two: a convolution kernel optimized to conflate halftone dots with each other as to restore exactly the information possible.  (Of course, such a kernel would have to be individually tuned for the orientation/spacing of the dots.)Literally painting over frequency peaks in the FFT with black circles I imagine would be pretty lossy, and not entirely rid yourself of the pattern (since you're making a new pattern with your dots).  Indeed, in the animation, the image does get darker as circles are added, and some of the pattern is still visible.Perhaps using a blur tool to blur out the peaks in the FFT would serve to maintain original image tone, and further reduce patterning?