### Godot version

v4.0.alpha11.official [afdae67cc] (when the feature was added) and newer (still happens in RC6)

### System information

Windows 10

### Issue description

I'm unable to handle when the user sets an exported node property (this feature: https://github.com/godotengine/godot/pull/62185) from a setter in the editor (tool script).

I don't know if this is intentional or not, but for me it was unexpected. I had to revert back to `NodePath`s.

### Steps to reproduce

- Create a scene
- Attach a script to a node
- Add `@tool` at the top of the script
- Export a node property with a setter, preferably with an easy way to see if it executed:
	```
	@export var node:Node:
		set (mod_value):
			prints("node", mod_value)
			node = mod_value
	```
- Save, close and open again the scene to make sure the tool script is running in the editor
- Set the node property from the inspector
- Observe that the setter didn't run

The setter is not bypassed when running the game, it is only bypassed in the editor.

We can compare with a `NodePath` property, which setter is not bypassed regardless if it is in the editor or in game runtime.

### Minimal reproduction project

[nodepath.zip](https://github.com/godotengine/godot/files/10840411/nodepath.zip)

if (i == amount) {
				PanelContainer *pc = memnew(PanelContainer);
				property_vbox->add_child(pc);
				pc->add_theme_style_override(SNAME("panel"), get_theme_stylebox(SNAME("DictionaryAddItem"), SNAME("EditorStyles")));
				add_vbox = memnew(VBoxContainer);
				pc->add_child(add_vbox);
			}
			prop->set_object_and_property(object.ptr(), prop_name);
			int change_index = 0;
			if (i < amount) {
				String cs = key.get_construct_string();
				prop->set_label(key.get_construct_string());
				prop->set_tooltip_text(cs);
				change_index = i + offset;
			} else if (i == amount) {
				prop->set_label(TTR("New Key:"));
				change_index = -1;
			} else if (i == amount + 1) {
				prop->set_label(TTR("New Value:"));
				change_index = -2;
			}
			prop->set_selectable(false);
			prop->connect("property_changed", callable_mp(this, &EditorPropertyDictionary::_property_changed));
			prop->connect("object_id_selected", callable_mp(this, &EditorPropertyDictionary::_object_id_selected));
			HBoxContainer *hbox = memnew(HBoxContainer);
			if (add_vbox) {
				add_vbox->add_child(hbox);
			} else {
				property_vbox->add_child(hbox);
			}