I was curious about who would win: the wither or the dragon, so I conducted a few experiments. These were all conducted in Vanilla Minecraft, in creative mode.

Experiment 1: 1 on 1

The Mr Withers seemed more interested in Endermen than dragons. Who am I to judge?

Experiment 2: They say no (ender)man is on this island

I used a command block to instantly kill every Enderman that spawned. Mr Withers chased Dr Agon around but was ineffective. Dr Agon was refuelling regularly, which seemed a bit unfair on Mr Withers!

Experiment 3: I pity the fuel

I destroyed the dragon feeding stations, then let Mr Withers do his work. Just between you and me, he is not the sharpest axe in the chest. Not that Dr Agon was any better. He just flew around thinking dragonny thoughts while Mr Withers chased him casually tossing his missiles and not really achieving much at all.

Experiment 4: Withering Heights

Maybe one Mr Withers was not enough? I made 9 friends for Mr Withers, to see what would happen. The posse leapt straight into action, dutifully chasing Dr Agon around as he completed his circuit with robotic disdain. Sadly, internal disputes within the party caused them to turn on one another, and they proceeded to slowly but surely destroy themselves. In the time it took the final Mr Withers to die in the agony of failure, they had only managed to reduce Dr Agon's health by about 1/7.

The verdict

The wither may be hard for humans to defeat, but he's no more than a mosquito to the dragon!

Bonus round: What the blazes?

I used a repeating command block to make an enormous mess of Blazes. They started chasing Mr Withers up in the air. He was completely unfazed happily floated up and away, picking them off like fish in a barrel. At some point I could no longer see them, only their smoke clouds, and at around 850 metres high he killed off the last one and returned home to roost.