### 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)

save_mode = true;
}
void EditorPropertyPath::_notification(int p_what) {
	switch (p_what) {
		case NOTIFICATION_ENTER_TREE:
		case NOTIFICATION_THEME_CHANGED: {
			path_edit->set_icon(get_theme_icon(SNAME("Folder"), SNAME("EditorIcons")));
		} break;
	}
}
void EditorPropertyPath::_path_focus_exited() {
	_path_selected(path->get_text());
}
void EditorPropertyPath::_drop_data_fw(const Point2 &p_point, const Variant &p_data, Control *p_from) {
	const Dictionary drag_data = p_data;
	if (!drag_data.has("type")) {
		return;
	}
	if (String(drag_data["type"]) != "files") {
		return;
	}
	const Vector<String> filesPaths = drag_data["files"];
	if (filesPaths.size() == 0) {
		return;
	}
	emit_changed(get_edited_property(), filesPaths[0]);
	update_property();
}
bool EditorPropertyPath::_can_drop_data_fw(const Point2 &p_point, const Variant &p_data, Control *p_from) const {