Wouldn't a consumer drone lose sight of the deer if it runs into the woods? (Greene County is at the northern edge of the Catskill Mountains and is mostly rural and wooded). You might want to contact the Albany County Sheriff's office. Albany County Sheriff has drones with thermal imaging cameras that they use to locate missing hikers. This would be a good training opportunity for them. I don't know if the Greene County Sheriff has drones, but I would start with them and if they don't have the drones, they could make the ask of the Albany County Sheriff.
I had posted this as a PM, but no reason it can't be public