I can think of four* reasons why the dragon would want to pilot a fighter plane rather than fly in its dragon form, with the three reasons leading to the most important fourth one.
One reason would be similar to why people watch movies or play games; to do something novel that they normally cannot do. Rather than flying with their own body, they can control the plane to do maneuvers, like a sort of game.
Aerodynamics, Speed, Stamina
Apart from that, a dragon’s normal flying form is probably the last thing anyone would think of when considering aerodynamics. A dragon, assuming without the use of any magics, would probably fly no faster than a bird when in the air. To fly quickly at a reasonable speed would also quickly consume the dragon’s stamina or magic as well.
Compare that to a fighter plane, which can easily hit subsonic or supersonic speeds, and consumes no stamina to run, only the measliest of fossil fuels. It would also be much more agile and potentially deadly, being able to fire missiles rather than using up the dragon’s magic powers.
Not to mention, in the event of emergencies, the dragon still has its dragon form as backup, with a full tank of stamina and magic.
Organic vs Mass-Produced
Finally, considering the existence of fighter planes in such an age, a dragon in dragon form would be nothing more than a sitting duck. Due to its slow speed, it would practically be a stationary target for fighter planes. Even if it has magic, if it is swarmed by fifty or five hundred fighter planes it would probably eventually run out of magic and be forced into a sorry state.
Even though the secret civilization is the only one possessing such a technology, who knows if that will be so in the future? By forming an alliance, the dragon gets to try out the new technology and find its weaknesses, giving itself an advantage should other civilization adopt similar technologies.
This problem mainly exists because the dragon is a single, organic entity, meaning it has a limited amount of magic and stamina. The fighter plane is the culmination of the efforts of countless human beings, and can be mass produced. As a single organic entity, the dragon cannot withstand the might of an entire civilization, and thus, needs to make use of their efforts to improve its own efficiency.
King of the Skies
Inspired by the discussion in the comments
The most important reason, since the dragon itself has probably considered the prior three. Assuming the dragon is extremely powerful, and capable of magic (due to shape-shifting), giving it a fighter plane would be the equivalent of making it the King of the Skies. In dragon form, there are weaknesses that allow enough enemies to swarm it and tire it out, but what if that same dragon was in a fighter plane?
Assuming a dragon, to fly, would need to use magic to reduce drag, function as an air-brake, and alter its ability to withstand high-G turns and maneuvers, can't these things be applied to the fighter plane too? You would then have a fighter plane that is much quicker than its peers, capable of doing inhuman maneuvers mid-air, and piloted by a dragon with faster/better/stronger reflexes and an insane amount of aerial combat experience. Even if they sent hundreds of fighter planes to go after such an opponent, would anyone be capable of shooting down or even catching up to such a fighter plane?
Using the human technology would allow the dragon to remain the king of the skies, no matter how much human technologies progress.
In conclusion, the fighter plane is probably novel to the dragon, and also allows it to fly faster, save energy, and be competitive in a tough job market. By making use of the highly advanced fighter plane, with its experience and magic, the dragon would remain the dominator of the skies.