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Did you buy both the DLC and game through GOG? I have no experience with gog, but AFAIK they have a game client which you can use to install their games and DLCs. And then you can start eu4 from that client which should give you the eu4 branded version of the paradox launcher. In the launcher there is a DLC tab in which you can enable/disable DLCs.
 
Yes, if you bought a non-GOG version of that DLC, it will never work with the GoG base game.

If that is the case, get a refund for that DLC from where you bought it, and buy from GoG or any resellers they may have instead.