Another Dev took these steps to verify that it is, in fact, impossible for Spine to set that behavior in Unity. These are the steps he took:
1.) Check out the cat source and open in Spine4.
2.) Export settings, one export folder with loop checked, and one folder with loop unchecked.
3.) Verification using Bitwise compare of exact match of exports.
Conclusion: The playback button in the Spine editor is not an export setting and Spine exports have no such value. When you import spine files they are loop agnostic and any such setting is determined via C# code. Your developers are responsible for loop control via code.
Can you confirm that this analysis makes sense? Dev A is still not entirely convinced.