### 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)

String new_name = rename_dialog_text->get_text().strip_edges();
	if (new_name.length() == 0) {
		EditorNode::get_singleton()->show_warning(TTR("No name provided."));
		return;
	} else if (new_name.contains("/") || new_name.contains("\\") || new_name.contains(":")) {
		EditorNode::get_singleton()->show_warning(TTR("Name contains invalid characters."));
		return;
	}
	names.set(renamed_layer_index, new_name);
	tooltips.set(renamed_layer_index, new_name + "\n" + vformat(TTR("Bit %d, value %d"), renamed_layer_index, 1 << renamed_layer_index));
	emit_signal(SNAME("rename_confirmed"), renamed_layer_index, new_name);
}
EditorPropertyLayersGrid::EditorPropertyLayersGrid() {
	rename_dialog = memnew(ConfirmationDialog);
	VBoxContainer *rename_dialog_vb = memnew(VBoxContainer);
	rename_dialog->add_child(rename_dialog_vb);
	rename_dialog_text = memnew(LineEdit);
	rename_dialog_vb->add_margin_child(TTR("Name:"), rename_dialog_text);
	rename_dialog->set_ok_button_text(TTR("Rename"));
	add_child(rename_dialog);
	rename_dialog->register_text_enter(rename_dialog_text);
	rename_dialog->connect("confirmed", callable_mp(this, &EditorPropertyLayersGrid::_rename_operation_confirm));
	layer_rename = memnew(PopupMenu);
	layer_rename->add_item(TTR("Rename layer"), 0);
	add_child(layer_rename);
	layer_rename->connect("id_pressed", callable_mp(this, &EditorPropertyLayersGrid::_rename_pressed));
}
Size2 EditorPropertyLayersGrid::get_grid_size() const {