"Battle Simulation" is sort of vague. How about "Avatar Battles"? Or maybe "Trainer House", since that's the place in GSC which does this.
I'm inclined toward the use of Trainer House. I would go with that, including having CAL there by default.
I'm guessing the inclusion of being able to choose what Trainer Type of your Gender you want to be like the official games ave isn't something that's going to be included with this though, as it is something that's arguably unnecessary (Though the entire feature is actually technically unnecessary)