The rules in Battle Royale are far more unique and removed from what you will be traditionally accustomed to in Pokemon games. No single battles, no double battles here.
Here's how the Battle Royale plays!
Lv.50 and Species Clause are in effect
Similar to the Battle Tree, all Pokemon are capped off at Level 50. If Pokemon are above Level 50, the game will reduce them down to Level 50 for the match.
Additionally, you bring a team of three Pokemon, like in Battle Tree Singles, and you cannot bring duplicate Pokemon or have them hold the exact same hold items.
The definition of a duplicate Pokemon is tied to Pokedex number, which means you cannot bring two formes of one Pokemon. For example, Zygarde 10% and Zygarde-Complete are classified as duplicates.
Additionally, all Legendary and Mythical Pokemon are banned. The Zygarde example presented above is illegal to use in the Battle Royale.
Its a four-way free-for-all
You are pitted up against three other NPCs for a four-on-four match, with each bringing out one Pokemon. Any Pokemon can smash any other Pokemon in the face, and so it's not unusual to see NPCs gang up on someone--usually you, or otherwise killsteal you away from victory.
Moves that affect both opponents in a Double Battle will affect the whole field here, but have power cut by 50%, which heavily decreases incentive to use them. Moves such as Follow Me and Spotlight will still work in full effect here, and offer chances for interesting and ruthless strategies.
The match ends when one trainer is defeated
This is not a last-man-standing death match! The Battle Royale match ends as soon as one trainer is fully defeated. Victory is tallied by who has the most number of Pokemon left, and the number of KOs everyone was able to get.
So if one trainer is fully defeated, and at that time someone on the field has suffered no losses, they are the winner if everyone else has suffered at least one loss, unless they managed to defeat nobody.
This means that NPCs kill-stealing you can easily cost you a victory, and it's not uncommon for NPCs to gang up on you and take you down.
If there is a tie, then the ultimate victory goes to whoever has the most numerical hit points left in their surviving Pokemon totaled.
This obviously has huge implications. You don't need to survive the fight, rather, you want to blitz first, get KOs, and protect your Pokemon from being targeted in response. This adds a whole new layer of strategy and can make the Battle Royale really fun!
Battle Royale Ranks
There are multiple ranks with Battle Royale you can ascend to and win in. Obviously, winning in higher tiers nets you more BP, but be careful, they're also much more difficult. The ranks follow as such: Normal Rank, Super Rank, Hyper Rank, and finally Master Rank.
Note that you can win BP even if you lose, and if you win you will win upwards of 5 BP per match. If you are in the Battle Royale purely to farm BP, sticking to Hyper Rank and Super Rank is a safe bet.