Mirror PC stats, abilities and numbers to give them a challenge. If you have a fighter, wizard and rogue group, and you want to put them up against monsters, you'll need to set them against something tanky, something ranged, and something that can hit hard and fast.
Alternately, you could design special monsters like a Colossal Scorpion. Give the torso the tanky stats, and make each pincer a rogue type damage dealer. In addition, the stinger could be considered limited ranged (30'/60'). This would give the scorpion 3 distinct parts that can be targetted, as well as a main body that will kill the whole thing.
You can apply this to any set of monster enemies as well. An Ogre, axe throwing orc, and orc shaman would be a good trio to set up against PC's if you balance their stats according to the PC stats.
The problem you're going to be facing with trying to balance this is something you've already pointed out, the CR ratings don't mesh with single fights. It will take a good amount of experimentation to get a feel for it, but if you start with PC stats and incorporate those into monsters, you should be fine using that as a baseline.
Numbers version of level 6 characters:
Fighter - 65 HP - 2 handed weapon
Wizard - 32 HP - evocation
Rogue - 40 HP - Whip, throwing knives
Colossal Scorpion +7 to hit (Mutli-attack, this creature attacks with two claw attacks and a stinger attack)
Legendary Resistance 1 - Once per encounter, this creature can turn a failed save into a success
Regenerate - As a bonus action, the scorpion can sacrifice 40 Torso HP to fully regenerate a destroyed claw or stinger. That claw or stinger only becomes available for use on the creatures next turn.
Torso - 150 HP - damage resistant non-magical
Left Claw - 40 HP - Pincer attack 1d10 + 3
Right Claw - 40 HP - Crushing Grip 1d10 +3, auto-grapples
Stinger - 40 HP - Acidic spray, ranged 30'/60' 1d8 + 4 acid damage
If you kill the claws and stinger, the scorpion dies. If you kill the torso, the scorpion dies.
That's just one suggested battle to keep things interesting. The main point is that a battle like that enables a monster to take as many actions as your players can, and it doesn't destroy the creatures ability to remain a challenge.
If you break the enemy down into trios as well, consider giving them similar abilities to player characters. This will ensure the challenge rating is appropriate. Try to avoid making the mistake of jacking up their AC, to hit bonus, HP and resistances. Keep it consistent with what the PC's have to maintain the arena feel to the tournament.