I am an arborist and can comment on some of this. First, as for climbing the tree yourself, I knew a man who wanted to trim his own tree. The branches were about 15 feet off the ground. He fell, broke his neck and died a few days later. He left his new bride in a very difficult situation. It is easy to get hurt climbing trees. How far out on the limb is your drone? That is also a complicating factor.
How would an arborist get up your tree? Generally, I would use a sack filled with shot that is attached to a thin, but very strong cord. I have a very special sling shot that I launch the sack over very sturdy branches. I retrieve the shot sack on the ground after it has been shot up and over the branch and I remove the shot sack, and tie on my climbing rope. I use the think cord to pull my climbing rope over the sturdy branches and back down to me. Then I clip ascenders onto one end of the rope and climb the rope to get to the location where the rope went over the branch. Do I need to climb higher? There are other challenging methods to continue upwards. Do I need to climb far out to the edge of small limbs to reach the drone? If so, I would fix my rope much higher than the drone and rappel down some distance to the level of the drone and then limb walk outwards to the drone. The rope fixed from above will allow me to pendulum and walk outwards, but it's not for the feint of heart. You can see the sling shot that I used by searching for the Notch Big Shot Set (2x 4' Poles) on Amazon. However, my rope also costs nearly $200 and the ascenders are also several hundred dollars in total, add on the price of the harness and the flip line... it's not cheap.
Keep in mind I have a cable covered with nylon rope, called a flip line, that I can solidly anchor into the tree at important places. I have other attachments that I can use to secure myself. Sometimes the branches that I shoot the rope over break and I have to know how to back that up and maintain my safety. It is a complicated process that is impossible to completely explain here.
IF you can find a sling shot and shoot a line over the limb in a location away from the trunk, you might be able to shake the limb and dislodge the drone without climbing the tree. That would be the safest option for you. A professional sling shot and gear would cost well over $100. I live in California. If you want to send me an airline ticket, I'll come and rescue it for you. That would be an expensive solution since you are in NC.
