I have modified set the following .ini files for Skyrim Special Edition:

...\Documents\My Games\Skyrim Special Edition GOG\Skyrim.ini
...\GOG Galaxy\Skyrim Anniversary Edition\Skyrim_Default.ini
...\GOG Galaxy\Skyrim Anniversary Edition\Skyrim_Default_en.ini

to:

[Display]
bAllowScreenshot=0

which was initially set to 1, but I still get .png screenshots printed to ...\GOG Galaxy\Skyrim Anniversary Edition\ and I still get the "ScreenShot: File 'ScreenShot0.png' created." pop-up in game.

Is the GOG copy of Skyrim SE configured differently than Fallout 3 or Fallout: New Vegas, where bAllowScreenShot worked to disable to in-game screenshot functionality? Any way to disable it here in Skyrim, as I would rather use my third party software without the imprinted pop-up and lag?

I'm not familiar with this problem and I don't even own Skyrim, so I could be definitely wrong, but I suspect that Galaxy could be the cause of this behavior.
Have you tried disabling the Galaxy overlay or launching the game by its own executable, without Galaxy running?