Very simple .po / .mo language switcher

Hello,

I’m just setting up a new worpdress website, that basically will containt blog articles and a WooCommerce shop. I need to make the site trilingual: Italian, English and Latin (yes, Latin! It’s about ancient languages.)
There is a pletora of plugins to manage multilingual sites but a lot of them feature automatic or AI translation, thing that I wan’t to avoid at all cost.
I already managed to translate the .po and .mo files of the theme to both Italian and Latin (the theme is originally in English) and I don’t need to translate the blog post (I’ll make them all trilingual, despite the navigation language).
So, what I’m looking for is a simple switcher that allow to change the translation of the theme (via .po/.mo files), of the menu and of WooCommerce.
Something extremely easy, even because I can’t waste time coding or modify settings.
Is there anything like that?

Thanks!

The page I need help with: [log in to see the link]