Another possible option would be to use a SPDT switch, to switch between the two joysticks:
Wire both Joysticks as "Player 1".
Rewire the ground (Common) connection for the joystick directional switches, to the center pole of the switch.
Wire the other switch contact to the ground connections for each joystick's (Up, Down, Left, Right) common switch connections.
That way, you can flip the switch for the current player, and only that joystick is "electrically" active, due to the "ground" completing the switch circuit.
As a bonus, use a DPDT switch, and wire an LED to the other pole (half) of the switch.
Mount the LED near the active joystick, as a visual cue, indicating which player's joystick will control the game character.
Just a suggestion.

- John