Actually, you can change aircraft in the trainers, but it requires using a console command.
Once you've started the trainer, press '~' to open the console. Then use the console command "loadvehicle <vehicle name>" to load the desired aircraft.
Example: "loadvehicle p-51 mustang".
You can type "help loadvehicle" for more information about that command.
Close the console by pressing '~' again or by clicking the 'X' in its upper right corner.