### Godot version

v4.1.1.stable.official [bd6af8e0e]

### System information

Godot v4.1.1.stable - Debian GNU/Linux trixie/sid trixie - Vulkan (Compatibility) - Mesa Intel(R) HD Graphics 520 (SKL GT2) () - Intel(R) Core(TM) i7-6500U CPU @ 2.50GHz (4 Threads)

### Issue description

After right clicking in thet "Filter Nodes" menu in the scene tree dock, the sub-menu is populated with a "Search" icon.

If I later switch themes in the same session, this icon is never updated to match the new theme.

![image](https://github.com/godotengine/godot/assets/660911/14336bb3-12ca-48c8-b030-8ada6ac27ec6)


### Steps to reproduce

1. Right click in the "Filter Nodes" menu in scene tree dock
2. Switch themes
3. Right click again in the "Filter Nodes" menu

Expected result: Icon has updated to match the theme I am currently using
Actual result: Icon is still using the theme I launched Godot with

### Minimal reproduction project

N/A
button_custom->set_text(TTR("Other Node"));
			button_custom->set_icon(get_theme_icon(SNAME("Add"), SNAME("EditorIcons")));
			button_custom->connect("pressed", callable_mp(this, &SceneTreeDock::_tool_selected).bind(TOOL_NEW, false));
			button_clipboard = memnew(Button);
			node_shortcuts->add_child(button_clipboard);
			button_clipboard->set_text(TTR("Paste From Clipboard"));
			button_clipboard->set_icon(get_theme_icon(SNAME("ActionPaste"), SNAME("EditorIcons")));
			button_clipboard->connect("pressed", callable_mp(this, &SceneTreeDock::_tool_selected).bind(TOOL_PASTE, false));
			_update_create_root_dialog();
		} break;
		case NOTIFICATION_ENTER_TREE: {
			clear_inherit_confirm->connect("confirmed", callable_mp(this, &SceneTreeDock::_tool_selected).bind(TOOL_SCENE_CLEAR_INHERITANCE_CONFIRM, false));
			scene_tree->set_auto_expand_selected(EDITOR_GET("docks/scene_tree/auto_expand_to_selected"), false);
		} break;
		case NOTIFICATION_EXIT_TREE: {
			clear_inherit_confirm->disconnect("confirmed", callable_mp(this, &SceneTreeDock::_tool_selected));
		} break;
		case EditorSettings::NOTIFICATION_EDITOR_SETTINGS_CHANGED: {
			scene_tree->set_auto_expand_selected(EDITOR_GET("docks/scene_tree/auto_expand_to_selected"), false);
		} break;
		case NOTIFICATION_THEME_CHANGED: {
			button_add->set_icon(get_theme_icon(SNAME("Add"), SNAME("EditorIcons")));
			button_instance->set_icon(get_theme_icon(SNAME("Instance"), SNAME("EditorIcons")));
			button_create_script->set_icon(get_theme_icon(SNAME("ScriptCreate"), SNAME("EditorIcons")));
			button_detach_script->set_icon(get_theme_icon(SNAME("ScriptRemove"), SNAME("EditorIcons")));
			button_tree_menu->set_icon(get_theme_icon(SNAME("GuiTabMenuHl"), SNAME("EditorIcons")));
			filter->set_right_icon(get_theme_icon(SNAME("Search"), SNAME("EditorIcons")));
			
			if (button_2d) {
				button_2d->set_icon(get_theme_icon(SNAME("Node2D"), SNAME("EditorIcons")));