Translation of Wordpress menu items when setting the locale programmatically

I have English and Deutch languages, WordPress 6.0. I programmatically set the language using switch_to_locale('de_DE'); (depends on URL).

add_action( 'plugin_loaded', 'set_lang');
function set_lang() {
    switch_to_locale('de_DE');
} 

Everything translates except menu items.

Menu items are translated if I set ?lang=de in the URL or execute $REQUEST['lang'] = 'de'; in hook plugins_loaded

This solution works but does not look correct. Tell me, please, how to programmatically set the language more correctly so that the menu is also translated.

I use Max Mega Menu and WP Multilang for Translation. I tried to use another template and menu plugin but it doesn't change language.

I tried to use add_filter('locale', .. ) but it doesn't help.

I tried to use switch_to_locale() in another hooks but it doesn`t help.

Thank you very much for your attention, I will be glad to solve my problem! Hope this helps more than just me.