Installing galaxy shouldn't affect anything. It should detect your current installation of NMS and just add it to galaxy and as far as I know it shouldn't affect your save file. It may automatically download any patches to update your game to the latest version (personally I have this turned off).

It isn't just a GOG problem. My friends and I wound up getting the PS4 version instead, and we still have trouble connecting to the game's servers...

Probably because the galaxy and manual installs are at different version numbers at the moment. Galaxy has a newer build. So if your were to override the galaxy version with the manual patch it could stuff things up and it was just warning you of that.

If you install galaxy you should be able to go to NMS in your library, click on more-manage installation-import folder and select your manual installation of NMS. It'll then hopefully scan the folder, figure out what version it is, and let you update it to the latest build.