### 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)

class EditorSpinSlider;
class PropertySelector;
class SceneTreeDialog;
class TextEdit;
class TextureButton;
class EditorPropertyNil : public EditorProperty {
	GDCLASS(EditorPropertyNil, EditorProperty);
	LineEdit *text = nullptr;
public:
	virtual void update_property() override;
	EditorPropertyNil();
};
class EditorPropertyText : public EditorProperty {
	GDCLASS(EditorPropertyText, EditorProperty);
	LineEdit *text = nullptr;
	bool updating = false;
	bool string_name = false;
	void _text_changed(const String &p_string);
	void _text_submitted(const String &p_string);
protected:
	virtual void _set_read_only(bool p_read_only) override;
	static void _bind_methods();
public:
	void set_string_name(bool p_enabled);
	virtual void update_property() override;
	void set_placeholder(const String &p_string);
	void set_secret(bool p_enabled);
	EditorPropertyText();
};
class EditorPropertyMultilineText : public EditorProperty {