Trainers are in set places. E.g. Floor 10 battle no. 3 (for the third badge) is where Torus can be met. The door you pick gives you a chance for encountering the trainer. No message = low chance, white message = medium chance, red message = high chance.
The trainer isn’t behind a certain one of the three doors, it’s a chance each time you select the door that you will meet the trainer there. I.e. To meet the trainer you should pick a red door then if you don’t meet them you have to retreat and re-enter the same door until you do. If you want to avoid a trainer you’ve got to pick a door with the lowest chance and try your luck. If you happen to meet the trainer anyway then it’s too late to retreat, you will lose all your badges if you do.
I made a map of where all the trainers are if you want to take a look: Showdown Tower map