Hi.
Im not sure at 100% but you dont need to use GameState for this. Theres also normally more than one ay to achieve most of the thing over there
In my point of View you should can do it all in the character controler BP. wich can contain any number of variable you need to control wich one the
players can act in his turn.
For example you can just disable the input of player 2 then player 1 should play his moves and viceversa.
also its usefull to post some screen of the part you think is problematic sometimes its easier to track an posible solution if we see what approach you took.