Stellaris 3.4.2 here, offline installers, no Galaxy, no mods. All my owned extensions are activated as usual.
I'm playing on Linux and do not use the Paradox Launcher.

DLC is typically controlled by Paradox. It might be you are logged out, if you use the same account for both Steam and GOG there won't be an issue unless there is a version mismatch.

Now this DLC checking only matters if you are using the launcher. As long as all the flags are already set correctly then Playing without the launcher or galaxy should be fine.

To Whom May Concern,
seems like the issue is related to their frikkin' Paradox Launcher v2 which, for some reason, stopped considering itself as a launcher for DRM-free game.
As far as I can tell, DRM'd and DRM-free versions of Stellaris use same Launcher.
The only difference is in command line options it uses. Somehow, my CLI options were partially gone - and it triggered redundant and unjustified DRM check.
Take care, reinstall their Launcher if/when you'll have same issue and double-check shortcuts with which you're running the game.

I had this problem a few years back. It's a problem with GOG's Galaxy servers not properly recognizing that you have bought the DLC's. You need to contact GOG support to get it sorted out.

It's not a DRM issue because you can get around it by downloading the manual installers, but every time Galaxy patches your game the problem will come back.