I don't know about the "Republic class" or if there is such a thing, but the USS Republic was mentioned in 1 or 2 episodes of TOS, the one I remember is "Patterns of Force" which is about those two planets having that computer war. They say in that episode that the USS Republic visited the planet 50 years ago.......so fandom attributed it to being either a Deadalus-class, Baton-Roueg-class, or the Mann-class (which only 1 source claims, and that is the early 1980's publication "Spaceflight Chronology".) Ironically, the Baton class was an item in that same book, and the book said it was the Connie predecessor. The Daedalus came in later fanzine when those TOS production drawings were released, and Mastercom Data Center included it in their master ship recognition chart and named both the Daedalus and the Baton as the Connie predecessors.....completely ignoring the Mann-class.
However, some people think the Republic mentioned in that episode was neither a Baton, Daedalus, or Mann but was in fact a Connie, because in the TOS episode "Court Martial" we see the Republic's name and registry on display as it being docked at the base. However...........it has a low reg number, 1371, so some say it may not have been a Connie. But then, official word from Paramount was that it is a Connnie in that episode.
Confusing eh?