You’re very close, it seems that you just need to initialise your style instance when your module is loaded (your Get() call doesn’t do this).
The code you have to initialise your module is failing to compile due to the DEFINE_LOG_CATEGORY(LogGame) line. If you look in StrategyGame.h you’ll see that it has the following:
DECLARE_LOG_CATEGORY_EXTERN(LogGame, Log, All);
You can either add that to your own games header (FPS_Inventory.h), or just remove the DEFINE_LOG_CATEGORY(LogGame) line.