So, one thing we didn't talk about was the political ramification that'd arise from kingdom A using a dragon in battles.
Kingdom A seems to be able to make the dragon do just about anything they say. The Dragon can rebel to a degree and choose his own method of doing things, but he can't disobey a direct order (without punishment).
The most reliable way to determine if having a dragon would just make more enemies is to look at what nobles and the church are thinking.
A quick rundown on the dragon
Combat value: While nowhere near as malicious or powerful as your typical Smaug, the Dragon is obviously much more powerful than any individual human he may encounter. He usually isn't fighting on the front lines, as his ability to fly allows him to wreak havoc deep within enemy territory then get out fast. So, he usually does stuff like attacking supply lines and tracking enemy movement.
Obviously, the Dragon isn't invincible, he can still get exhausted and poison works on him, though usually to a lesser degree when compared to humans.
Social Standing: Despite that, the Dragon is considered to be a pet at best and a slave at worst. The only people who treat him like an actual person are his handlers, who themselves are nobles, but are still pretty low in the hierarchy.
Reputation outside of combat: While the Dragon indeed likes to appear regal and is actually pretty smart, he's too stubborn and confident for his own good. He also does not understand human social norms often talking down to people that are several rungs above him and sometimes he also ends up being accidentally lewd when he forgets he isn't supposed to sniff strangers, lick friends in the face, or preen them, even if those are perfectly acceptable for dragons.
The general opinion considers him to be a beast/monster that's ultimately below humans. His handlers consider him to be simply childish and the church thinks he's a demon from hell.
Reputation in combat: Though he's willing to spare just about anyone who surrenders (read: runs away), there are three big problems:
- If someone manages to make him angry enough (usually by threatening his loved ones), he won't care if they're a noble, or even a king and tear them to shreds.
- His breath weapon is concentrated sulfuric acid that's sprayed in a fine mist, while usually not lethal, it can quickly cause permanent eye damage, blindness and lasting respiratory problems. The dragon will use it when his life is in danger, regardless of the target.
- Though the Dragon has no incentive to and takes no pleasure from hurting humans, that's completely irrelevant to people who are afraid that kingdom A might send him on terror missions like a Red Wedding.
Other than the Dragon, the setting has no fantastical elements (as far as humans know, there's only one dragon) and is more or less a carbon copy of high medieval Europe.
How could kingdom A minimize the "bad press", using the Dragon would give them?