Wouter,
I don't mind.

Yes, everything was done in PixInsight.
For this one, I followed an unorthodox pipeline. Raw files were loaded with in-camera white balance and no back point correction. I substracted the bias from the frames, and then rescaled each channel by the maximum value (I calculated it for the longest frame, and used the same factors for all frames). To avoid SNR issues, I applied a mild TGVDenoise process to the linear data. Then, frames were merged manually, using PixelMath (I was having some artifacts with the HDRCombination tool). The equation I used was: $T*0.17*(1-$T)+$T*im2. The 0.17 factor comes from the time difference between 2 frames (I used more decimals). I worked over the largest exposition, converted to 64bits floating point to avoid losing any data due to rounding. The im2 image was updated each time, to add consecutively shorter frames, one at a time. In the end, I divided by the scaling factors to recover the white balance.
Regarding the alignment of the frames, I didn't do anything.

Please see my reply to Rob below for more details.
For the HDR compression, I also used an unorthodox method.

I combined results using a scale separation approach and the multiscale gradient domain compression tool I wrote a few years ago (with a much lower weight).
The scale separation approach bases its rationale in the same approach as the Homomorphic filter. In general, images are composed of illumination and reflection components. These are multiplicative effects. By taking the logarithm they become additive terms, easier to isolate. Whereas the Homomorphic filters models illumination as the low-frequency components, a multiscale approach uses spatial filters. Instead of wavelets, I used ACDNR to produce a blurred version, with a protected moon-corona boundary. By taking the difference with the log-image I was able to get both large scale (the blurred version) and small scale components. I processed each separately to compress the range of values in the corona/sky background, and to enhance the details (flares, corona, etc). Here wavelets and the Larson-Sekanina filters played a major part.
After the combination of the large and small scale components, I returned to the "linear" range by exponentiating, and then I sed Histogram and curves transforms to fine-tune the appearance. I also increased the color saturation for the solar flares only. Finally, since I got a better result for the appearance of the moon using the gradient domain compression tool, I used a mask to put a stronger contribution from that result in the final image.
Rob,
There where two dim stars. Unfortunately, they were also very soft, and not visible without heavy stretching (and visible only in the longest exposition). I couldn't also use them for PSF estimation, as the PSF was also different between frames.
If I were to merge frames from different sets, I would have used them to align, but this was not the case here. Thanks to the fact I had a decent polar alignment, I didn't have the need to align these particular frames. Due to the wind and maybe the periodic error, I had some movement in other sets. FFTregistration did a fair job with those, although the moon moves a little bit. The Canon 80D camera allows doing 7 bracketed frames automatically, so the movement of the moon if minimized a lot.
I was having some posterization effects in my earliest attempts. I figured that the reason was a slight mistmatch between the linear factors that was calculated by HDRCombination and the theoretical ones, and also the fact that it uses a monocrome mask. To minimize these problems, I did it manually as described to Wouter. Instead of a strong mask, I preferred to use the same data to mask the incorporation of the shorter frames. Since they had been already denoised, it did not degrade the data in a significant way.
To deal with the "pink stars" effect I introduced the scaling factors in the pipeline. But, the problem was not only that. The saturated region varied in size between the channels. A single monochrome mask cannot do the job here.
Thank you both for your compliments!