I think the available public loos in Harborne may changed since you worked at the library, I don't recognize the ones you mention. The only "official" public loos I'm aware of are in the Coop (petrol station) further up the High St past the library away from the city, in the M&S and one of those unreliable 10p city council pods loos outside Iceland.
Other than that (esp. after the shops close), pubs are your only bet (as is the case in most of Birmingham at any time of day outside shopping areas, and the city centre after the shops close).
In the Green Man they're right by the front door. I much prefer the system in German where you could just put 50c down on the counter of virtually any pub or restaurant and use their facilities without having to either sneak, or embarrassingly ask and risk refusal. read more