### 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)

paginator->update(page_index, max_page);
		paginator->set_visible(max_page > 0);
		int amount = MIN(size - offset, page_length);
		for (int i = 0; i < amount; i++) {
			bool reorder_is_from_current_page = reorder_from_index / page_length == page_index;
			if (reorder_is_from_current_page && i == reorder_from_index % page_length) {
				
				continue;
			}
			if (!reorder_is_from_current_page && i == reorder_to_index % page_length) {
				
				
				
				continue;
			}
			HBoxContainer *hbox = memnew(HBoxContainer);
			property_vbox->add_child(hbox);
			Button *reorder_button = memnew(Button);
			reorder_button->set_icon(get_theme_icon(SNAME("TripleBar"), SNAME("EditorIcons")));
			reorder_button->set_default_cursor_shape(Control::CURSOR_MOVE);
			reorder_button->set_disabled(is_read_only());
			reorder_button->connect("gui_input", callable_mp(this, &EditorPropertyArray::_reorder_button_gui_input));
			reorder_button->connect("button_down", callable_mp(this, &EditorPropertyArray::_reorder_button_down).bind(i + offset));
			reorder_button->connect("button_up", callable_mp(this, &EditorPropertyArray::_reorder_button_up));
			hbox->add_child(reorder_button);
			String prop_name = "indices/" + itos(i + offset);
			EditorProperty *prop = nullptr;
			Variant value = array.get(i + offset);
			Variant::Type value_type = value.get_type();
			if (value_type == Variant::NIL && subtype != Variant::NIL) {