Hi Gunnar,
first, I would like to welcome you here in our forum known as the best Crossfire knowledge base in and around Germany.
Second, I was face with a similar problem because the second key was broken. I ordered a new one from a Chrysler/Jeep dealer not too far away from my location. It tooks almost 3 weeks until arrived and it's well described here in the forum, how to adopt the key to your vehicle.
I did it as follows and it worked perfect:
When the key arrives at the dealeship, it must be initialized to the vehice before the keyless entry will function. A technician or the customer (you) must use the new key to start the engine 1 to 5 times to complete the initialization. After initialization the keyless entry will function!
I think, this is the best way to handle this issue 
Have a good time with your Crossfire - unfortunately it can't fly although you bought it from a pilot…
Best wishes,
Alois 