See Syncthing’s “Ignoring Files” page: https://docs.syncthing.net/users/ignoring.html
In a nutshell, you’ll be most interested in the “negated pattern” syntax. You just need two lines in your ignore pattern:
!/some/folder/on/my/pc/game.dat
**
First line is the path to the individual file you want to sync (relative to the root of your Syncthing folder). Second line is a wildcard to ignore everything else including any subfolders.
As for whether using a different program is better depends on what and/or where the game data file needs to be synced to.