### Godot version

4.0.stable

### System information

Windows 10, Forward+, GTX 1070 driver 528.49

### Issue description

When trying to edit the handle mode of a bezier cruve key from the inspector tab, the mode doesn't change. It works fine when right clicking on the key and setting the handle mode from here though. 

https://user-images.githubusercontent.com/24484962/222379540-29d71b6d-ff0c-4a21-9905-9750d0581fda.mp4



### Steps to reproduce

On an animation track using a bezier curve, select a key and try to change the handle mode from the inspector tab

### Minimal reproduction project

N/A
GDCLASS(AnimationTrackEditPlugin, RefCounted);
public:
	virtual AnimationTrackEdit *create_value_track_edit(Object *p_object, Variant::Type p_type, const String &p_property, PropertyHint p_hint, const String &p_hint_string, int p_usage);
	virtual AnimationTrackEdit *create_audio_track_edit();
	virtual AnimationTrackEdit *create_animation_track_edit(Object *p_object);
};
class AnimationTrackKeyEdit;
class AnimationMultiTrackKeyEdit;
class AnimationBezierTrackEdit;
class AnimationTrackEditGroup : public Control {
	GDCLASS(AnimationTrackEditGroup, Control);
	Ref<Texture2D> icon;
	String node_name;
	NodePath node;
	Node *root = nullptr;
	AnimationTimelineEdit *timeline = nullptr;
	void _zoom_changed();
protected:
	void _notification(int p_what);
	virtual void gui_input(const Ref<InputEvent> &p_event) override;
public:
	void set_type_and_name(const Ref<Texture2D> &p_type, const String &p_name, const NodePath &p_node);
	virtual Size2 get_minimum_size() const override;
	void set_timeline(AnimationTimelineEdit *p_timeline);
	void set_root(Node *p_root);
	AnimationTrackEditGroup();
};
class AnimationTrackEditor : public VBoxContainer {