A fascinating project, I like it. Certainly a challenge. My not-so-experienced approach would likely be to hammer out the equirectangular projection of the planet with a grey/black region, or even green screen like shooting a movie, where the missing piece is. Then transfer that into an Orthographic globe projection. Take that back into Photoshop or GIMP for the artsy fartsy elimination of the chunk of planet. Portraying a missing section in equirectangular... sounds like a challenge, glad you are the one doing it! But I look forward to the finished product, should be cool.