Stefan, what a great-looking Cocoon!
Excellent job capturing and processing those dusty areas - I like the way you were able to reveal it in much the same way one often sees well-done images of the dusty dark nebula around the Iris. Great star colors too!
Your a-bin link was just the jpg, so I couldn't find my way to the technical card. Would love to know the acquisition specs.
But no I did not have this problem, as my data was not nearly so deep as yours.
I only had 3 hours of RGB and 5 hours of stripped Ha, if I remember right. So, stretching more in order to try to get the brown dust just made a wreck of the whole image with excessive noise.
In the end I boosted Cocoon detail (and redness) as described, which may not have been the most correct blending, but it did look nice. I also maintained the busy star field, one because I thought it looked neat and didn't warrant excessive shrinking, but also because if I couldn't pull out the brown dust, I could at least show how the dark nebula was blocking the stars behind it.
For the "problem" you describe in your image, you might want to try the SS module in sort of a "backwards" mode. I downloaded your jpg, converted it to tiff, and tested it in ST with a non-linear open. Go to SS and hit the saturate mode. Then, pull the saturation slider way way down, to like 5 to 15 percent maybe. Also, bump the gamma, to 1.0 or greater. This should desaturate mostly just the super structures, meaning your Cocoon, and help reveal extra detail in doing so, as well as decreasing its relative obnoxiousness if you feel it to be too strong here. You will lose a little bit of star color, but play with the adjustments how you like and maybe you can find a good balance.
One possibility, which I did not test, is to set the image up for this beforehand. Say, go into color and actually boost the saturation in contemplation of the desaturation to come in SS module. That may not work, I'm really not sure. We would need to know exactly what SS considers to be "super structures," and if it varies in degree (such as, the Cocoon is clearly major super structure and so gets maximum SS effects, but the star halos/coloring are a lower-degree super structure and so get less of the SS effects).
