Thanks for the suggestion, I gave it a shot.
First, I uninstalled just the high resolution patch and tried running the game. I got the same results as before, the game minimized every few seconds.
I then decided to uninstall everything and reinstall Fallout and try it without any of the additional mods or patches, just straight as it comes from GOG. I'm still having the problem, though, the only difference being that it minimizes every 10 seconds or so, instead of every three.
So it looks to me like one of the mods or patches is possibly aggravating the problem, but the root of the problem is that something is wrong with the way the game from GOG and my computer are interacting.