Custom Skybox Priority Over All Brushes
I have a skybox made for cs:s used in my map, but when I use it, when I get into the room that has it showing, every brush becomes invisible until I turn back probably because of the different leaf. I have not built a 3D skybox yet.
That's because of the parameter $ignorez 1... just remove that line from all the .VMT files 
Check also what shader are your skybox .VMT files using, if it's "Sky", change it to "UnlitGeneric".
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[/spoiler]Now it does the same thing with the pink missing texture checkerboard.
Where are your files located?
How many files have you got for this skybox?
Also, can you post the content of at least 1 of the .VMT files?
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[/spoiler]The content is correct except for the "sky"... Change it for "UnlitGeneric" as I told you before.
About the path, I've got all mine into portal 2/portal2/materials/skybox... not into the dlc2 folder; to be honest I'm not sure if this could make a difference becasue I never tried. Try putting them under portal 2/portal2/materials/skybox and check...
Check also what did you put in hammer into the Map Menu and Map Properties. For that sky you should put only:
sky37
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No difference. Both are supported by gameinfo.txt.