### 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)

if (resource_picker) {
		memdelete(resource_picker);
		resource_picker = nullptr;
	}
	if (p_path == "script" && p_base_type == "Script" && Object::cast_to<Node>(p_object)) {
		EditorScriptPicker *script_picker = memnew(EditorScriptPicker);
		script_picker->set_script_owner(Object::cast_to<Node>(p_object));
		resource_picker = script_picker;
	} else if (p_path == "shader" && p_base_type == "Shader" && Object::cast_to<ShaderMaterial>(p_object)) {
		EditorShaderPicker *shader_picker = memnew(EditorShaderPicker);
		shader_picker->set_edited_material(Object::cast_to<ShaderMaterial>(p_object));
		resource_picker = shader_picker;
		connect(SNAME("ready"), callable_mp(this, &EditorPropertyResource::_update_preferred_shader));
	} else if (p_base_type == "AudioStream") {
		EditorAudioStreamPicker *astream_picker = memnew(EditorAudioStreamPicker);
		resource_picker = astream_picker;
	} else {
		resource_picker = memnew(EditorResourcePicker);
	}
	resource_picker->set_base_type(p_base_type);
	resource_picker->set_editable(true);
	resource_picker->set_h_size_flags(SIZE_EXPAND_FILL);
	add_child(resource_picker);
	resource_picker->connect("resource_selected", callable_mp(this, &EditorPropertyResource::_resource_selected));
	resource_picker->connect("resource_changed", callable_mp(this, &EditorPropertyResource::_resource_changed));
	for (int i = 0; i < resource_picker->get_child_count(); i++) {
		Button *b = Object::cast_to<Button>(resource_picker->get_child(i));
		if (b) {
			add_focusable(b);
		}