Hmm, I was asuming the name was known after some time, at least in the Watch who have a sharp eye on what is going on in the south. Scar asks the PC on the bridge but I always thought it's more of a courtesy he lets the PC reveal their name in a question - BG city authorities should know who they are expecting or we have a situation like Greywolf and the PC in Nashkel with Oublek.
My assumption was: BG city Watch / Flaming Fist knows about CHARNAME clearing out the mines in Nashkel.
The soldier in the prison heard the tower commander and Scar talking about CHARNAME. Not every soldier knows about it. Sorry for giving a creepy feeling that everyone knows CHARNAME, it was supposed to be Scar+Tower Commander+the soldiers who were guarding the meeting of which one of them drew his own conclusions that he wants CHARNAME's help.
If I take this out the main reason why the Commander hands the prison pass to CHARNAME would be gone. I see your point but I'd need a good other suggestion to work around this. I could make the Commander's acknowledgement dependent on whether the PC is actually already working for Scar or at least revealed themselves to Scar. That would make more sense maybe?
The inn is on the first floor of the main tower. The map calls the whole tower Balduran's Seatower. I'm open to suggestions how to clearify the locality.