Thanks to support, I have an embarrassingly simple solution for my case at least: (1) run the GOG Fallout 2 installer as an administrator; and (2) run the game from the GOG shortcut, but as an administrator. In both cases, this can be accomplished by right-clicking on the relevant icon and choosing the "Run as Administrator option".
Since presumably you'll do step #2 more than once, you can also edit the shortcut so that it runs as administrator by default by opening up properties and checking the "Run this program as an administrator" box on the "Compatibility" tab.