Trouble setting the object reference in a property drawer

I've got a wrapper class that holds a serialzed named key and a data object. I'd like to select from static instances of the wrapper in a dropdown based on the key. My assignment of those references doesn't seem to be working correctly. What's the proper way to assign a reference to a new object in a property drawer?

Here's some example code:

     [Serializable]
     class Wrapper
     {
         public string name;
         private Data data;
     }
 
     class Data
     {
         public string foo;
         public int bar;
     }
 
     class Drawer : PropertyDrawer
     {
         public override void OnGUI(Rect position, SerializedProperty property, GUIContent label)
         {
             int index = 3; //from dropdown
 
             Undo.RegisterCompleteObjectUndo(property.serializedObject.targetObject, "Setting wrapper");
             fieldInfo.SetValue(property.serializedObject.targetObject, StaticClass.listOfWrappers[index]);
             property.serializedObject.ApplyModifiedProperties();
         }
     }
 
     class StaticClass
     {
         public static List<Wrapper> listOfWrappers = new List<Wrapper>() { new Wrapper() { name = "foo" }, new Wrapper() { name = "bar" } };
     }

This seems to almost work. If I have two instances of a component with a wrapper field and I set one to foo and one to bar, save, and change scenes then come back they are both set to bar.

I tried making the wrapper a UnityEngine.Object and then setting property.objectReferenceValue. But that didn't seem to stick at all. I'd prefer for it not to be a unity engine object as I don't think I get any benefit from that.

Any ideas what I'm doing wrong?