Battle Arena FAQ:

Basic Definition: The Battle Arena is the arena facility. You have to progress 7 3v3 battles. However after 3 turns in any battle, a judging takes place to decide who wins immediately. 3 Factors, "Mind", "Skill", and "Body" are taken into account. If you lose, you faint immediately. A tie causes both Pokemon to faint. You also can not switch or change your order of Pokemon, which is the real difficulty of this facility

Difficulty: 6th of 7 on easiness scale (Very Hard)

Team: Latios @ Lum Berry: Calm Mind, Psychic, Dragon Claw, Thunderbolt
Metagross @ Choice Band: Meteor Mash, Earthquake, Shadow Ball, Explosion
Swampert @ Leftovers: Earthquake, Surf, Ice Beam, Counter

Manipulation: None. You could technically double frame perfect clear to manip the first fight, but it's both ridiculous and not worth it to do so.

The 3 factors work like so:
Mind: Simply how many times you use a move that does damage (Has Power), even if it misses. Exception: Fake out gives minus 1, as does Protect/Detect/Endure, so you can't cheap it out.
Skill: This refers to using effective damaging moves. So normal hit is +1, Super Effective hit is +2, Not Very Effective hit is -1, Missing is -2, Failing because of ability is -3. There are quite a few exceptions, but none of them really matter.
Body: Refers to HP. Whoever lost the lower percentage of HP during the 3 turns wins.

So basically you just pummel down to win anyway. Exceptions are like Wobbuffet, where you win by just using Calm Mind 3 times. The tricky part about this facility is you can't switch OR change your order, so if you come up against a bad order you can just lose instantly. E.g. if a Houndoom comes out and managed to kill Latios, I'm forced to send in Metagross, which then also dies, instead of being able to use Swampert.