Strictly speaking there's nothing wrong grammatically with this. What is unusual about it, and what makes it grate slightly to (at least this) native speaker, is that there is a clash of styles: "got" in this context is slightly informal (though entirely normal in conversation), whereas "most tired" is rather formal/old fashioned.
Either would be fine alone, in the right context, but putting the two together makes for a strange mix. To fix it, while keeping the meaning the same:
If you want to be formal, replacing "got" with "grew" is better than "became", which to me feels slightly clunky in this context.
If you want to go the other way use "very" in place of "most".