### 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)

switch (p_what) {
		case NOTIFICATION_THEME_CHANGED:
		case NOTIFICATION_ENTER_TREE: {
			change_type->clear();
			for (int i = 0; i < Variant::VARIANT_MAX; i++) {
				if (i == Variant::CALLABLE || i == Variant::SIGNAL || i == Variant::RID) {
					
					continue;
				}
				String type = Variant::get_type_name(Variant::Type(i));
				change_type->add_icon_item(get_theme_icon(type, SNAME("EditorIcons")), type, i);
			}
			change_type->add_separator();
			change_type->add_icon_item(get_theme_icon(SNAME("Remove"), SNAME("EditorIcons")), TTR("Remove Item"), Variant::VARIANT_MAX);
			if (button_add_item) {
				button_add_item->set_icon(get_theme_icon(SNAME("Add"), SNAME("EditorIcons")));
			}
		} break;
		case NOTIFICATION_DRAG_BEGIN: {
			if (is_visible_in_tree()) {
				if (_is_drop_valid(get_viewport()->gui_get_drag_data())) {