Just some rambling on my part since I'm not there to help and see exactly what you got going on there.
First, don't use 2 inch screws. All you need do is measure the thickness of the wall where the door is to get a good idea of how thick your walls are. The better rigs will be thicker then the cheaper ones.
Second, you have studs in the back wall which are more then likely 16 inch centers. Now the question is are they horizontal or vertical I can not say. Sometimes you can see the stud showing on the outside skin when there is a morning dew on the rig.
Third the idea of screwing a board to the studs then the swing arm to the board is a good suggestion. That should be strong enough to hold your TV.
Forth, since you got a Class A I hope the studs would be welded steel or aluminum. If they are wood I would sell the rig and upgrade before you start to have future problems.
