>>31539568Solution:
1. Make a new save (back up your old one, of course) and play until you get access to the menu, and get yourself a Game Sync ID.
2. Export it using SDF or SVDT to get your "main" file.
3. Open it in a hex editor and copy 8 bytes starting from 0x14008 to 0x1400F.
4. Open your other save that you want to change the Game ID for with a hex editor, and paste over 0x14008 to 0x1400F with the new Game Sync ID you made.
5. Use PKHeX to open your old "main" file, and use Export SAV to fix checksums.
6. Import your save again with SaveDataFiler or SVDT (erase the secure value if necessary).
7. Now you should be able to use your new Game Sync ID on the PGL and sync in-game.
I had wanted to figure out where exactly the game decided you had an ID or not, but after a couple hours of tinkering, I had no luck. But now we know 0x14008-0x1400F controls your Game Sync ID. It's basically as easy as starting a new game, getting an ID, and transplanting said new ID into your old game.