I got back to this issue today and I solved it like a boss thanks to the Zanto plugin.
Just to recap, I was trying to translate the front-end of only one site from a multisite setup with the least possible hassle while leaving every other site unchanged.
All I did was copy the general WP language files into wp-content/languages and the translated theme language files into a languages folder inside the (parent) theme. I got those from the translated WP package I downloaded on the slovak WP website. Both are named sk_SK.po (and .mo).
By activating the plugin on the translated site, I could easily switch the language and that’s all I needed. Great plugin!