Ok, so… Unfortunately, I wasn’t able to find a solution.
No matter what I tried, no matter how much I googled - nothing works.
So, instead I decided to go for the old fashioned “fake it til you make it” approach.
I ditched volumetric fog altogether. Adjusted exponential fog, added some fake lightshafts as particle emitters, tweaked HDRI in photoshop to match the outside “fogginess” with the inside more or less, re-did the lighting and other small tweaks…
Now I have a skybox that looks good.
P.S. A thing to note: do not do the “translucent skybox material” if you have windows in your scene. It will cause translucency sorting issues. Just regular “opaque unlit” is fine and adjust exponential fog cutoff distance.