### 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)

setup(layer_type);
}
void EditorPropertyLayers::_bind_methods() {
}
EditorPropertyLayers::EditorPropertyLayers() {
	HBoxContainer *hb = memnew(HBoxContainer);
	hb->set_clip_contents(true);
	add_child(hb);
	grid = memnew(EditorPropertyLayersGrid);
	grid->connect("flag_changed", callable_mp(this, &EditorPropertyLayers::_grid_changed));
	grid->connect("rename_confirmed", callable_mp(this, &EditorPropertyLayers::set_layer_name));
	grid->set_h_size_flags(SIZE_EXPAND_FILL);
	hb->add_child(grid);
	button = memnew(TextureButton);
	button->set_stretch_mode(TextureButton::STRETCH_KEEP_CENTERED);
	button->set_toggle_mode(true);
	button->connect("pressed", callable_mp(this, &EditorPropertyLayers::_button_pressed));
	hb->add_child(button);
	set_bottom_editor(hb);
	layers = memnew(PopupMenu);
	add_child(layers);
	layers->set_hide_on_checkable_item_selection(false);
	layers->connect("id_pressed", callable_mp(this, &EditorPropertyLayers::_menu_pressed));
	layers->connect("popup_hide", callable_mp((BaseButton *)button, &BaseButton::set_pressed).bind(false));
	EditorNode::get_singleton()->connect("project_settings_changed", callable_mp(this, &EditorPropertyLayers::_refresh_names));
}
void EditorPropertyInteger::_set_read_only(bool p_read_only) {