### 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)

clear();
		ERR_FAIL_V(err);
	}
	err = _parse_connections(scene, scene, name_map, variant_map, node_map, nodepath_map);
	if (err) {
		clear();
		ERR_FAIL_V(err);
	}
	names.resize(name_map.size());
	for (const KeyValue<StringName, int> &E : name_map) {
		names.write[E.value] = E.key;
	}
	variants.resize(variant_map.size());
	for (const KeyValue<Variant, int> &E : variant_map) {
		int idx = E.value;
		variants.write[idx] = E.key;
	}
	node_paths.resize(nodepath_map.size());
	for (const KeyValue<Node *, int> &E : nodepath_map) {
		node_paths.write[E.value] = scene->get_path_to(E.key);
	}
	if (Engine::get_singleton()->is_editor_hint()) {
		
		for (const KeyValue<Node *, int> &E : node_map) {
			node_path_cache[scene->get_path_to(E.key)] = E.value;
		}
	}
	return OK;
}
void SceneState::set_path(const String &p_path) {