I found a way to make that pop-up go away, and it doesn't involve updating the video drivers or modifying the game's executable file.

Turns out that if you simply rename the "nvapi64.dll" file in the System32 folder, the game launches and no pop-up ever shows up. It can be renamed to literally anything, apparently. Hell, that's actually the name I gave it: "anything.dll," and everything works without a hitch.

Now, given the original filename starts with the letters "nv" I'm thinking that this file is specific to nVidia video cards, so it probably won't work with AMD cards. Still, it's nice to know that there's a way of getting rid of this annoying pop-up, especially considering that the pop-up in question is produced by the game, and not by Windows, despite the fact the pop-up is made to appear like an official Windows thing.

So now I can play the game comfortably without being harassed by the idiots at EA for not having the "recommended" driver. Good stuff.

EDIT: Aaaand it turns out that though this helps the game run, it also causes problems if you don't rename it back. Thankfully, someone over on the nVidia forums has figured out a way to make this process much simpler by use of a batch file: https://forums.geforce.com/default/topic/962633/pc-games/bypassing-the-driver-version-check-of-a-game

All you have to do with this is run it as admin and it'll do all the stuff you need automatically, and then when you're done playing you can simply change it back.