Answer
From IFSAPP10 onward, changing the database connection string isn't done through the installer/reconfigure process at all — it's done through the IFS Middleware Server Admin Console instead, where a dedicated field lets you update it (the IFSSYS system-user password field becomes visible, and the save button enables, once you actually change the connection string). This is one of several configuration items that moved from the installer into the MWS Admin Console starting with Apps10 — if something that used to be configurable during installation on earlier releases seems to be missing, checking the MWS Admin Console for it is generally the right next step.