I would not get the DA 50-200mm for plane spotting, and even 300mm is going to be too short. But if you don't want to go for a manual focus 400mm f/5.6, then either the Sigma 70-300mm or the Pentax DA 55-300mm would be my choices.
Back in the day I bought the Sigma because the DA 55-300mm hadn't been released yet. Today I would go with the Pentax.
If you do consider a 400mm f/5.6, search for the Tokina
SD version or the Sigma
APO.
Here is a shot with an autofocus Tokina 400mm f/5.6 (which is over your budget, but shows what a 400mm can do). This is a full frame cropped on the side to 4x5 ratio: