Ok couldn't help myself. So I decided to sim the monk all the way, including ki use and everything, to see how effective it truly is against the dragon 1v1. No subclasses, straight up.
NOTE: Technically in a 1v1, the dragon doesn't get all its legendary actions. But I think that's been changed in a lot of recent monsters so I'm assuming that.
I simmed with the monk going 3 different styles. Again no mercy monk this time, no real subclass features included.
Always: Superior Defense put on round 1 and whenever it runs out (assuming there ki left)
Sim 1: Patient Defense as a ki using bonus action. So 2 attacks per round + dodge + temp HP. Result: Dragon has 246 HP left.
Sim 2: Dodge as an action + flurry using Ki. So 3 attacks per round + dodge + no temp HP. Result: Dragon has 296 HP left.
Sim 3: Attack as an action + flurry using Ki. So 5 attacks per round. Result: Dragon has 320 HP left.
So it appears that "patience" is the best play for the monk against such an offense heavy opponent. But yeah the monk isn't going to beat a dragon 1v1.