After trying and failing several times in the past with image mask based methods, I decided to finally take the plunge on the vortex shader. I'm pretty pleased with the results. There are two v3 cloud layers for the main storm, with a couple other clouds for the surroundings.
I did have some strange things going on with lighting. I found that when I brought the sun elevation down, the clouds in the center of the storm (which have the density increased using a simple shape shader) seem to disappear (see StormA). When I raise the sun up just a few degrees, they come back very strongly (StormB).
Each image took around 5 hours each to render, so I ended up just masking them together in Affinity, using StormB to provide some extra density to StormA, (and added some stars
)