An Azure relational database service.
Is that an application in Azure App Service, or something on-prem? If the application is hosted in a single place, what's the point with having three different instances? Do the users also connect to the database by other means?
My naïve thinking is that when the user connects, you look up the user in a database (which could be a small Azure SQL Database), and you get the connection option for that user from that database. This assumes that you know all your users beforehand. If you don't, you would need to find a way to trace where they are coming from. That may be trivial or impossible depending on the situation. Maybe you need to ask them the first time they connect. (The reason I say that it may be trivial or impossible is because I don't know what users we are talking about. Users in a corporate organisation? Users from the public internet? Something else?)