Change in Story Mode

Its not an easy answer, but it is possible using functionalities other than story mode.
I recommend using a more standard method like UFE.StartPlayerVersusCpu() and code your logic behind it. Story Mode is set in a way of auto-progress where a map system would allow you to replay previous matches at will.

Take a look at this page: http://www.ufe3d.com/doku.php/code

I also took the liberty and added a new overload to the method UFE._StartStoryModeBattle:
_StartStoryModeBattle(int groupNumber, float fadeTime = 0)

Even though StartPlayerVersusCpu() will do the job, using the method above will allow you to set which group number you want to start the story mode from. Might be useful if you want to use some of the Story Mode functionalities.

Expect this functionality to be available on the next update. If you have the Source version however you can just add this method to UFE.cs:

public static void _StartStoryModeBattle(int groupNumber, float fadeTime = 0){
   UFE.storyMode.currentGroup = groupNumber;
   _StartStoryModeBattle(fadeTime);
}

Humm. If you want to skip the character selection all together, use this script:

public class StartCustomMatch : MonoBehaviour
{
    public UFE3D.CharacterInfo player1CharacterFile;
    public UFE3D.CharacterInfo player2CharacterFile;

    void Start()
    {
        UFE.config.player1Character = player1CharacterFile;
        UFE.config.player2Character = player2CharacterFile;
        UFE.SetCPU(1, false); // Set to true so player 1 is controlled by CPU
        UFE.SetCPU(2, true); // Set to false so player 2 is controlled by player

        UFE.StartLoadingBattleScreen();
        //UFE.StartGame(); // Use StartGame if you want to skip the loading screen
    }
}

Alternatively, you can load UFE in individual scenes and use the 'Start Game Immediately' toggle in combination with the Player 1/2 Character fields (that can also be changed programmatically)