Though I am no expert ....
I think you should try what you did with the mouse and the cabinet ... but with the tree. As a person who had a dog (Leeo) who was a huge ratter and high prey drive guy ... the only thing I could do was let him have his way with his target ... to keep him from "screaming" from the excitement of the hunt! lol!
This would calm him enough to re-direct him to what I wanted him to do.
I think you should try what you did with the mouse and the cabinet ... but with the tree. As a person who had a dog (Leeo) who was a huge ratter and high prey drive guy ... the only thing I could do was let him have his way with his target ... to keep him from "screaming" from the excitement of the hunt! lol!