Alright, the Smash Wiki had listed down all the references of the Ring Fit Adventure spirit battles from the event, here they are:
Ring Fit Trainees - The Wii Fit Trainers obviously represent the trainees themselves given the two also originate from a series of fitness games. With Golden Plains probably representing the beginning levels of Ring Fit Adventure, the coins that pop up in the stage reference the collectible rings that can be obtained in throughout various levels. The condition where the Trainers suddenly breathe fire references how the Trainees' hair looks when synced together with their companion Ring, which also represents the player's heart rate when playing.
Dragaux - Dragaux's upper body strength is reflected with Incineroar, while the stage of choice reference his home stadium that appears at the end of each world in Ring Fit Adventure, where the player must beat him in a boss battle.
Ring - With this battle, the enemy fighter being equipped with the Gust Bellows references the Ring's ability to inhale or bust out air much like a vaccum cleaner. The invisible Bayonetta primarily attacking with smash attacks references the Ring's ability to summon holographic limbs for certain moves in RFA. The detail of how Bayonetta's Wicked Weaves open up circular portals that resemble rings to unleash the giant fists and feet of Madame Butterfly might also reference the opening part of RFA where Dragaux is released from inside of Ring and sets off the game. Lastly, Garden of Hope resembles the first stage set in Beginnia where the Trainee first meets Ring.