The phrase "bark up the wrong tree" comes from hunting dogs that chase an animal up a tree but end up barking at the base of the wrong one, while the real prey has escaped elsewhere. Over time this picture turned into an idiom for pursuing a false lead or aiming your effort at the wrong target because of a wrong assumption.
Option 4, "Make the wrong choice," captures this sense best. The person is not being careless or dishonest, they simply picked the wrong direction based on faulty information, just like the dog barking at the empty tree.
Option 3 is close but wrong in tone. "Purposely make an error" suggests the mistake is deliberate, while the idiom always describes an honest, unintentional mistake. Options 1 and 2 take the words literally, "skin of another animal" and "behave like a dog," which misses the figurative meaning entirely.